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Theorem aomclem8 42292
Description: Lemma for dfac11 42293. Perform variable substitutions. This is the most we can say without invoking regularity. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Hypotheses
Ref Expression
aomclem8.a (𝜑𝐴 ∈ On)
aomclem8.y (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
Assertion
Ref Expression
aomclem8 (𝜑 → ∃𝑏 𝑏 We (𝑅1𝐴))
Distinct variable groups:   𝜑,𝑏   𝐴,𝑎,𝑏   𝑦,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑦,𝑎)   𝐴(𝑦)

Proof of Theorem aomclem8
Dummy variables 𝑐 𝑑 𝑒 𝑓 𝑔 𝑖 𝑗 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elequ2 2113 . . . . . . 7 ( = 𝑏 → (𝑖𝑖𝑏))
2 elequ2 2113 . . . . . . . 8 (𝑔 = 𝑐 → (𝑖𝑔𝑖𝑐))
32notbid 318 . . . . . . 7 (𝑔 = 𝑐 → (¬ 𝑖𝑔 ↔ ¬ 𝑖𝑐))
41, 3bi2anan9r 637 . . . . . 6 ((𝑔 = 𝑐 = 𝑏) → ((𝑖 ∧ ¬ 𝑖𝑔) ↔ (𝑖𝑏 ∧ ¬ 𝑖𝑐)))
5 elequ2 2113 . . . . . . . . 9 (𝑔 = 𝑐 → (𝑗𝑔𝑗𝑐))
6 elequ2 2113 . . . . . . . . 9 ( = 𝑏 → (𝑗𝑗𝑏))
75, 6bi2bian9 638 . . . . . . . 8 ((𝑔 = 𝑐 = 𝑏) → ((𝑗𝑔𝑗) ↔ (𝑗𝑐𝑗𝑏)))
87imbi2d 340 . . . . . . 7 ((𝑔 = 𝑐 = 𝑏) → ((𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)) ↔ (𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏))))
98ralbidv 3169 . . . . . 6 ((𝑔 = 𝑐 = 𝑏) → (∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)) ↔ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏))))
104, 9anbi12d 630 . . . . 5 ((𝑔 = 𝑐 = 𝑏) → (((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗))) ↔ ((𝑖𝑏 ∧ ¬ 𝑖𝑐) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏)))))
1110rexbidv 3170 . . . 4 ((𝑔 = 𝑐 = 𝑏) → (∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗))) ↔ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖𝑏 ∧ ¬ 𝑖𝑐) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏)))))
12 elequ1 2105 . . . . . . 7 (𝑖 = 𝑑 → (𝑖𝑏𝑑𝑏))
13 elequ1 2105 . . . . . . . 8 (𝑖 = 𝑑 → (𝑖𝑐𝑑𝑐))
1413notbid 318 . . . . . . 7 (𝑖 = 𝑑 → (¬ 𝑖𝑐 ↔ ¬ 𝑑𝑐))
1512, 14anbi12d 630 . . . . . 6 (𝑖 = 𝑑 → ((𝑖𝑏 ∧ ¬ 𝑖𝑐) ↔ (𝑑𝑏 ∧ ¬ 𝑑𝑐)))
16 breq2 5142 . . . . . . . . 9 (𝑖 = 𝑑 → (𝑗(𝑒 dom 𝑒)𝑖𝑗(𝑒 dom 𝑒)𝑑))
1716imbi1d 341 . . . . . . . 8 (𝑖 = 𝑑 → ((𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏)) ↔ (𝑗(𝑒 dom 𝑒)𝑑 → (𝑗𝑐𝑗𝑏))))
1817ralbidv 3169 . . . . . . 7 (𝑖 = 𝑑 → (∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏)) ↔ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑑 → (𝑗𝑐𝑗𝑏))))
19 breq1 5141 . . . . . . . . 9 (𝑗 = 𝑓 → (𝑗(𝑒 dom 𝑒)𝑑𝑓(𝑒 dom 𝑒)𝑑))
20 elequ1 2105 . . . . . . . . . 10 (𝑗 = 𝑓 → (𝑗𝑐𝑓𝑐))
21 elequ1 2105 . . . . . . . . . 10 (𝑗 = 𝑓 → (𝑗𝑏𝑓𝑏))
2220, 21bibi12d 345 . . . . . . . . 9 (𝑗 = 𝑓 → ((𝑗𝑐𝑗𝑏) ↔ (𝑓𝑐𝑓𝑏)))
2319, 22imbi12d 344 . . . . . . . 8 (𝑗 = 𝑓 → ((𝑗(𝑒 dom 𝑒)𝑑 → (𝑗𝑐𝑗𝑏)) ↔ (𝑓(𝑒 dom 𝑒)𝑑 → (𝑓𝑐𝑓𝑏))))
2423cbvralvw 3226 . . . . . . 7 (∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑑 → (𝑗𝑐𝑗𝑏)) ↔ ∀𝑓 ∈ (𝑅1 dom 𝑒)(𝑓(𝑒 dom 𝑒)𝑑 → (𝑓𝑐𝑓𝑏)))
2518, 24bitrdi 287 . . . . . 6 (𝑖 = 𝑑 → (∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏)) ↔ ∀𝑓 ∈ (𝑅1 dom 𝑒)(𝑓(𝑒 dom 𝑒)𝑑 → (𝑓𝑐𝑓𝑏))))
2615, 25anbi12d 630 . . . . 5 (𝑖 = 𝑑 → (((𝑖𝑏 ∧ ¬ 𝑖𝑐) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏))) ↔ ((𝑑𝑏 ∧ ¬ 𝑑𝑐) ∧ ∀𝑓 ∈ (𝑅1 dom 𝑒)(𝑓(𝑒 dom 𝑒)𝑑 → (𝑓𝑐𝑓𝑏)))))
2726cbvrexvw 3227 . . . 4 (∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖𝑏 ∧ ¬ 𝑖𝑐) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏))) ↔ ∃𝑑 ∈ (𝑅1 dom 𝑒)((𝑑𝑏 ∧ ¬ 𝑑𝑐) ∧ ∀𝑓 ∈ (𝑅1 dom 𝑒)(𝑓(𝑒 dom 𝑒)𝑑 → (𝑓𝑐𝑓𝑏))))
2811, 27bitrdi 287 . . 3 ((𝑔 = 𝑐 = 𝑏) → (∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗))) ↔ ∃𝑑 ∈ (𝑅1 dom 𝑒)((𝑑𝑏 ∧ ¬ 𝑑𝑐) ∧ ∀𝑓 ∈ (𝑅1 dom 𝑒)(𝑓(𝑒 dom 𝑒)𝑑 → (𝑓𝑐𝑓𝑏)))))
2928cbvopabv 5211 . 2 {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))} = {⟨𝑐, 𝑏⟩ ∣ ∃𝑑 ∈ (𝑅1 dom 𝑒)((𝑑𝑏 ∧ ¬ 𝑑𝑐) ∧ ∀𝑓 ∈ (𝑅1 dom 𝑒)(𝑓(𝑒 dom 𝑒)𝑑 → (𝑓𝑐𝑓𝑏)))}
30 nfcv 2895 . . 3 𝑐sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))})
31 nfcv 2895 . . . 4 𝑔(𝑦𝑐)
32 nfcv 2895 . . . 4 𝑔(𝑅1‘dom 𝑒)
33 nfopab1 5208 . . . 4 𝑔{⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}
3431, 32, 33nfsup 9442 . . 3 𝑔sup((𝑦𝑐), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))})
35 fveq2 6881 . . . 4 (𝑔 = 𝑐 → (𝑦𝑔) = (𝑦𝑐))
3635supeq1d 9437 . . 3 (𝑔 = 𝑐 → sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}) = sup((𝑦𝑐), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))
3730, 34, 36cbvmpt 5249 . 2 (𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))})) = (𝑐 ∈ V ↦ sup((𝑦𝑐), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))
38 nfcv 2895 . . . 4 𝑐((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))
39 nffvmpt1 6892 . . . 4 𝑔((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑐))
40 rneq 5925 . . . . . 6 (𝑔 = 𝑐 → ran 𝑔 = ran 𝑐)
4140difeq2d 4114 . . . . 5 (𝑔 = 𝑐 → ((𝑅1‘dom 𝑒) ∖ ran 𝑔) = ((𝑅1‘dom 𝑒) ∖ ran 𝑐))
4241fveq2d 6885 . . . 4 (𝑔 = 𝑐 → ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)) = ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑐)))
4338, 39, 42cbvmpt 5249 . . 3 (𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))) = (𝑐 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑐)))
44 recseq 8369 . . 3 ((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))) = (𝑐 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑐))) → recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) = recs((𝑐 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑐)))))
4543, 44ax-mp 5 . 2 recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) = recs((𝑐 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑐))))
46 nfv 1909 . . 3 𝑐 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})
47 nfv 1909 . . 3 𝑏 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})
48 nfmpt1 5246 . . . . . . . 8 𝑔(𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))
4948nfrecs 8370 . . . . . . 7 𝑔recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))))
5049nfcnv 5868 . . . . . 6 𝑔recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))))
51 nfcv 2895 . . . . . 6 𝑔{𝑐}
5250, 51nfima 6057 . . . . 5 𝑔(recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐})
5352nfint 4950 . . . 4 𝑔 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐})
54 nfcv 2895 . . . . . 6 𝑔{𝑏}
5550, 54nfima 6057 . . . . 5 𝑔(recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})
5655nfint 4950 . . . 4 𝑔 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})
5753, 56nfel 2909 . . 3 𝑔 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})
58 nfcv 2895 . . . . . . . . 9 V
59 nfcv 2895 . . . . . . . . . . . 12 (𝑦𝑔)
60 nfcv 2895 . . . . . . . . . . . 12 (𝑅1‘dom 𝑒)
61 nfopab2 5209 . . . . . . . . . . . 12 {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}
6259, 60, 61nfsup 9442 . . . . . . . . . . 11 sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))})
6358, 62nfmpt 5245 . . . . . . . . . 10 (𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))
64 nfcv 2895 . . . . . . . . . 10 ((𝑅1‘dom 𝑒) ∖ ran 𝑔)
6563, 64nffv 6891 . . . . . . . . 9 ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))
6658, 65nfmpt 5245 . . . . . . . 8 (𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))
6766nfrecs 8370 . . . . . . 7 recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))))
6867nfcnv 5868 . . . . . 6 recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))))
69 nfcv 2895 . . . . . 6 {𝑐}
7068, 69nfima 6057 . . . . 5 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐})
7170nfint 4950 . . . 4 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐})
72 nfcv 2895 . . . . . 6 {𝑏}
7368, 72nfima 6057 . . . . 5 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})
7473nfint 4950 . . . 4 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})
7571, 74nfel 2909 . . 3 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})
76 sneq 4630 . . . . . 6 (𝑔 = 𝑐 → {𝑔} = {𝑐})
7776imaeq2d 6049 . . . . 5 (𝑔 = 𝑐 → (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) = (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}))
7877inteqd 4945 . . . 4 (𝑔 = 𝑐 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) = (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}))
79 sneq 4630 . . . . . 6 ( = 𝑏 → {} = {𝑏})
8079imaeq2d 6049 . . . . 5 ( = 𝑏 → (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {}) = (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏}))
8180inteqd 4945 . . . 4 ( = 𝑏 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {}) = (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏}))
82 eleq12 2815 . . . 4 (( (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) = (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}) ∧ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {}) = (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})) → ( (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {}) ↔ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})))
8378, 81, 82syl2an 595 . . 3 ((𝑔 = 𝑐 = 𝑏) → ( (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {}) ↔ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})))
8446, 47, 57, 75, 83cbvopab 5210 . 2 {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})} = {⟨𝑐, 𝑏⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})}
85 fveq2 6881 . . . . 5 (𝑔 = 𝑐 → (rank‘𝑔) = (rank‘𝑐))
86 fveq2 6881 . . . . 5 ( = 𝑏 → (rank‘) = (rank‘𝑏))
8785, 86breqan12d 5154 . . . 4 ((𝑔 = 𝑐 = 𝑏) → ((rank‘𝑔) E (rank‘) ↔ (rank‘𝑐) E (rank‘𝑏)))
8885, 86eqeqan12d 2738 . . . . 5 ((𝑔 = 𝑐 = 𝑏) → ((rank‘𝑔) = (rank‘) ↔ (rank‘𝑐) = (rank‘𝑏)))
89 simpl 482 . . . . . 6 ((𝑔 = 𝑐 = 𝑏) → 𝑔 = 𝑐)
90 suceq 6420 . . . . . . . . 9 ((rank‘𝑔) = (rank‘𝑐) → suc (rank‘𝑔) = suc (rank‘𝑐))
9185, 90syl 17 . . . . . . . 8 (𝑔 = 𝑐 → suc (rank‘𝑔) = suc (rank‘𝑐))
9291adantr 480 . . . . . . 7 ((𝑔 = 𝑐 = 𝑏) → suc (rank‘𝑔) = suc (rank‘𝑐))
9392fveq2d 6885 . . . . . 6 ((𝑔 = 𝑐 = 𝑏) → (𝑒‘suc (rank‘𝑔)) = (𝑒‘suc (rank‘𝑐)))
94 simpr 484 . . . . . 6 ((𝑔 = 𝑐 = 𝑏) → = 𝑏)
9589, 93, 94breq123d 5152 . . . . 5 ((𝑔 = 𝑐 = 𝑏) → (𝑔(𝑒‘suc (rank‘𝑔))𝑐(𝑒‘suc (rank‘𝑐))𝑏))
9688, 95anbi12d 630 . . . 4 ((𝑔 = 𝑐 = 𝑏) → (((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))) ↔ ((rank‘𝑐) = (rank‘𝑏) ∧ 𝑐(𝑒‘suc (rank‘𝑐))𝑏)))
9787, 96orbi12d 915 . . 3 ((𝑔 = 𝑐 = 𝑏) → (((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔)))) ↔ ((rank‘𝑐) E (rank‘𝑏) ∨ ((rank‘𝑐) = (rank‘𝑏) ∧ 𝑐(𝑒‘suc (rank‘𝑐))𝑏))))
9897cbvopabv 5211 . 2 {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))))} = {⟨𝑐, 𝑏⟩ ∣ ((rank‘𝑐) E (rank‘𝑏) ∨ ((rank‘𝑐) = (rank‘𝑏) ∧ 𝑐(𝑒‘suc (rank‘𝑐))𝑏))}
99 eqid 2724 . 2 (if(dom 𝑒 = dom 𝑒, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒))) = (if(dom 𝑒 = dom 𝑒, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒)))
100 dmeq 5893 . . . . . . 7 (𝑙 = 𝑒 → dom 𝑙 = dom 𝑒)
101100unieqd 4912 . . . . . . 7 (𝑙 = 𝑒 dom 𝑙 = dom 𝑒)
102100, 101eqeq12d 2740 . . . . . 6 (𝑙 = 𝑒 → (dom 𝑙 = dom 𝑙 ↔ dom 𝑒 = dom 𝑒))
103 fveq1 6880 . . . . . . . . . 10 (𝑙 = 𝑒 → (𝑙‘suc (rank‘𝑔)) = (𝑒‘suc (rank‘𝑔)))
104103breqd 5149 . . . . . . . . 9 (𝑙 = 𝑒 → (𝑔(𝑙‘suc (rank‘𝑔))𝑔(𝑒‘suc (rank‘𝑔))))
105104anbi2d 628 . . . . . . . 8 (𝑙 = 𝑒 → (((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑙‘suc (rank‘𝑔))) ↔ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔)))))
106105orbi2d 912 . . . . . . 7 (𝑙 = 𝑒 → (((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑙‘suc (rank‘𝑔)))) ↔ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))))))
107106opabbidv 5204 . . . . . 6 (𝑙 = 𝑒 → {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑙‘suc (rank‘𝑔))))} = {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))))})
108 eqidd 2725 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑒 → (𝑦𝑔) = (𝑦𝑔))
109100fveq2d 6885 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑒 → (𝑅1‘dom 𝑙) = (𝑅1‘dom 𝑒))
110101fveq2d 6885 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝑒 → (𝑅1 dom 𝑙) = (𝑅1 dom 𝑒))
111 id 22 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑒𝑙 = 𝑒)
112111, 101fveq12d 6888 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 = 𝑒 → (𝑙 dom 𝑙) = (𝑒 dom 𝑒))
113112breqd 5149 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 = 𝑒 → (𝑗(𝑙 dom 𝑙)𝑖𝑗(𝑒 dom 𝑒)𝑖))
114113imbi1d 341 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = 𝑒 → ((𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)) ↔ (𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗))))
115110, 114raleqbidv 3334 . . . . . . . . . . . . . . . . . . 19 (𝑙 = 𝑒 → (∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)) ↔ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗))))
116115anbi2d 628 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝑒 → (((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗))) ↔ ((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))))
117110, 116rexeqbidv 3335 . . . . . . . . . . . . . . . . 17 (𝑙 = 𝑒 → (∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗))) ↔ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))))
118117opabbidv 5204 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑒 → {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))} = {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))})
119108, 109, 118supeq123d 9441 . . . . . . . . . . . . . . 15 (𝑙 = 𝑒 → sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}) = sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))
120119mpteq2dv 5240 . . . . . . . . . . . . . 14 (𝑙 = 𝑒 → (𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))})) = (𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))})))
121109difeq1d 4113 . . . . . . . . . . . . . 14 (𝑙 = 𝑒 → ((𝑅1‘dom 𝑙) ∖ ran 𝑔) = ((𝑅1‘dom 𝑒) ∖ ran 𝑔))
122120, 121fveq12d 6888 . . . . . . . . . . . . 13 (𝑙 = 𝑒 → ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)) = ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))
123122mpteq2dv 5240 . . . . . . . . . . . 12 (𝑙 = 𝑒 → (𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔))) = (𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))))
124 recseq 8369 . . . . . . . . . . . 12 ((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔))) = (𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))) → recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) = recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))))
125123, 124syl 17 . . . . . . . . . . 11 (𝑙 = 𝑒 → recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) = recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))))
126125cnveqd 5865 . . . . . . . . . 10 (𝑙 = 𝑒recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) = recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))))
127126imaeq1d 6048 . . . . . . . . 9 (𝑙 = 𝑒 → (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) = (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}))
128127inteqd 4945 . . . . . . . 8 (𝑙 = 𝑒 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) = (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}))
129126imaeq1d 6048 . . . . . . . . 9 (𝑙 = 𝑒 → (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {}) = (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {}))
130129inteqd 4945 . . . . . . . 8 (𝑙 = 𝑒 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {}) = (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {}))
131128, 130eleq12d 2819 . . . . . . 7 (𝑙 = 𝑒 → ( (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {}) ↔ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})))
132131opabbidv 5204 . . . . . 6 (𝑙 = 𝑒 → {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {})} = {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})})
133102, 107, 132ifbieq12d 4548 . . . . 5 (𝑙 = 𝑒 → if(dom 𝑙 = dom 𝑙, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑙‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {})}) = if(dom 𝑒 = dom 𝑒, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})}))
134109sqxpeqd 5698 . . . . 5 (𝑙 = 𝑒 → ((𝑅1‘dom 𝑙) × (𝑅1‘dom 𝑙)) = ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒)))
135133, 134ineq12d 4205 . . . 4 (𝑙 = 𝑒 → (if(dom 𝑙 = dom 𝑙, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑙‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑙) × (𝑅1‘dom 𝑙))) = (if(dom 𝑒 = dom 𝑒, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒))))
136135cbvmptv 5251 . . 3 (𝑙 ∈ V ↦ (if(dom 𝑙 = dom 𝑙, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑙‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑙) × (𝑅1‘dom 𝑙)))) = (𝑒 ∈ V ↦ (if(dom 𝑒 = dom 𝑒, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒))))
137 recseq 8369 . . 3 ((𝑙 ∈ V ↦ (if(dom 𝑙 = dom 𝑙, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑙‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑙) × (𝑅1‘dom 𝑙)))) = (𝑒 ∈ V ↦ (if(dom 𝑒 = dom 𝑒, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒)))) → recs((𝑙 ∈ V ↦ (if(dom 𝑙 = dom 𝑙, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑙‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑙) × (𝑅1‘dom 𝑙))))) = recs((𝑒 ∈ V ↦ (if(dom 𝑒 = dom 𝑒, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒))))))
138136, 137ax-mp 5 . 2 recs((𝑙 ∈ V ↦ (if(dom 𝑙 = dom 𝑙, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑙‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑙) × (𝑅1‘dom 𝑙))))) = recs((𝑒 ∈ V ↦ (if(dom 𝑒 = dom 𝑒, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒)))))
139 aomclem8.a . 2 (𝜑𝐴 ∈ On)
140 aomclem8.y . . 3 (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
141 neeq1 2995 . . . . 5 (𝑎 = 𝑐 → (𝑎 ≠ ∅ ↔ 𝑐 ≠ ∅))
142 fveq2 6881 . . . . . 6 (𝑎 = 𝑐 → (𝑦𝑎) = (𝑦𝑐))
143 pweq 4608 . . . . . . . 8 (𝑎 = 𝑐 → 𝒫 𝑎 = 𝒫 𝑐)
144143ineq1d 4203 . . . . . . 7 (𝑎 = 𝑐 → (𝒫 𝑎 ∩ Fin) = (𝒫 𝑐 ∩ Fin))
145144difeq1d 4113 . . . . . 6 (𝑎 = 𝑐 → ((𝒫 𝑎 ∩ Fin) ∖ {∅}) = ((𝒫 𝑐 ∩ Fin) ∖ {∅}))
146142, 145eleq12d 2819 . . . . 5 (𝑎 = 𝑐 → ((𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅}) ↔ (𝑦𝑐) ∈ ((𝒫 𝑐 ∩ Fin) ∖ {∅})))
147141, 146imbi12d 344 . . . 4 (𝑎 = 𝑐 → ((𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})) ↔ (𝑐 ≠ ∅ → (𝑦𝑐) ∈ ((𝒫 𝑐 ∩ Fin) ∖ {∅}))))
148147cbvralvw 3226 . . 3 (∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})) ↔ ∀𝑐 ∈ 𝒫 (𝑅1𝐴)(𝑐 ≠ ∅ → (𝑦𝑐) ∈ ((𝒫 𝑐 ∩ Fin) ∖ {∅})))
149140, 148sylib 217 . 2 (𝜑 → ∀𝑐 ∈ 𝒫 (𝑅1𝐴)(𝑐 ≠ ∅ → (𝑦𝑐) ∈ ((𝒫 𝑐 ∩ Fin) ∖ {∅})))
15029, 37, 45, 84, 98, 99, 138, 139, 149aomclem7 42291 1 (𝜑 → ∃𝑏 𝑏 We (𝑅1𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 844   = wceq 1533  wex 1773  wcel 2098  wne 2932  wral 3053  wrex 3062  Vcvv 3466  cdif 3937  cin 3939  c0 4314  ifcif 4520  𝒫 cpw 4594  {csn 4620   cuni 4899   cint 4940   class class class wbr 5138  {copab 5200  cmpt 5221   E cep 5569   We wwe 5620   × cxp 5664  ccnv 5665  dom cdm 5666  ran crn 5667  cima 5669  Oncon0 6354  suc csuc 6356  cfv 6533  recscrecs 8365  Fincfn 8935  supcsup 9431  𝑅1cr1 9753  rankcrnk 9754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-2o 8462  df-map 8818  df-en 8936  df-fin 8939  df-sup 9433  df-r1 9755  df-rank 9756
This theorem is referenced by:  dfac11  42293
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