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Theorem aomclem8 39539
Description: Lemma for dfac11 39540. 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 2120 . . . . . . 7 ( = 𝑏 → (𝑖𝑖𝑏))
2 elequ2 2120 . . . . . . . 8 (𝑔 = 𝑐 → (𝑖𝑔𝑖𝑐))
32notbid 319 . . . . . . 7 (𝑔 = 𝑐 → (¬ 𝑖𝑔 ↔ ¬ 𝑖𝑐))
41, 3bi2anan9r 636 . . . . . 6 ((𝑔 = 𝑐 = 𝑏) → ((𝑖 ∧ ¬ 𝑖𝑔) ↔ (𝑖𝑏 ∧ ¬ 𝑖𝑐)))
5 elequ2 2120 . . . . . . . . 9 (𝑔 = 𝑐 → (𝑗𝑔𝑗𝑐))
6 elequ2 2120 . . . . . . . . 9 ( = 𝑏 → (𝑗𝑗𝑏))
75, 6bi2bian9 637 . . . . . . . 8 ((𝑔 = 𝑐 = 𝑏) → ((𝑗𝑔𝑗) ↔ (𝑗𝑐𝑗𝑏)))
87imbi2d 342 . . . . . . 7 ((𝑔 = 𝑐 = 𝑏) → ((𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)) ↔ (𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏))))
98ralbidv 3194 . . . . . 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 3294 . . . 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 2112 . . . . . . 7 (𝑖 = 𝑑 → (𝑖𝑏𝑑𝑏))
13 elequ1 2112 . . . . . . . 8 (𝑖 = 𝑑 → (𝑖𝑐𝑑𝑐))
1413notbid 319 . . . . . . 7 (𝑖 = 𝑑 → (¬ 𝑖𝑐 ↔ ¬ 𝑑𝑐))
1512, 14anbi12d 630 . . . . . 6 (𝑖 = 𝑑 → ((𝑖𝑏 ∧ ¬ 𝑖𝑐) ↔ (𝑑𝑏 ∧ ¬ 𝑑𝑐)))
16 breq2 5061 . . . . . . . . 9 (𝑖 = 𝑑 → (𝑗(𝑒 dom 𝑒)𝑖𝑗(𝑒 dom 𝑒)𝑑))
1716imbi1d 343 . . . . . . . 8 (𝑖 = 𝑑 → ((𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏)) ↔ (𝑗(𝑒 dom 𝑒)𝑑 → (𝑗𝑐𝑗𝑏))))
1817ralbidv 3194 . . . . . . 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 5060 . . . . . . . . 9 (𝑗 = 𝑓 → (𝑗(𝑒 dom 𝑒)𝑑𝑓(𝑒 dom 𝑒)𝑑))
20 elequ1 2112 . . . . . . . . . 10 (𝑗 = 𝑓 → (𝑗𝑐𝑓𝑐))
21 elequ1 2112 . . . . . . . . . 10 (𝑗 = 𝑓 → (𝑗𝑏𝑓𝑏))
2220, 21bibi12d 347 . . . . . . . . 9 (𝑗 = 𝑓 → ((𝑗𝑐𝑗𝑏) ↔ (𝑓𝑐𝑓𝑏)))
2319, 22imbi12d 346 . . . . . . . 8 (𝑗 = 𝑓 → ((𝑗(𝑒 dom 𝑒)𝑑 → (𝑗𝑐𝑗𝑏)) ↔ (𝑓(𝑒 dom 𝑒)𝑑 → (𝑓𝑐𝑓𝑏))))
2423cbvralvw 3447 . . . . . . 7 (∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑑 → (𝑗𝑐𝑗𝑏)) ↔ ∀𝑓 ∈ (𝑅1 dom 𝑒)(𝑓(𝑒 dom 𝑒)𝑑 → (𝑓𝑐𝑓𝑏)))
2518, 24syl6bb 288 . . . . . 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 3448 . . . 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, 27syl6bb 288 . . 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 5129 . 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 2974 . . 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 2974 . . . 4 𝑔(𝑦𝑐)
32 nfcv 2974 . . . 4 𝑔(𝑅1‘dom 𝑒)
33 nfopab1 5126 . . . 4 𝑔{⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}
3431, 32, 33nfsup 8903 . . 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 6663 . . . 4 (𝑔 = 𝑐 → (𝑦𝑔) = (𝑦𝑐))
3635supeq1d 8898 . . 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 5158 . 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 2974 . . . 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 6674 . . . 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 5799 . . . . . 6 (𝑔 = 𝑐 → ran 𝑔 = ran 𝑐)
4140difeq2d 4096 . . . . 5 (𝑔 = 𝑐 → ((𝑅1‘dom 𝑒) ∖ ran 𝑔) = ((𝑅1‘dom 𝑒) ∖ ran 𝑐))
4241fveq2d 6667 . . . 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 5158 . . 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 7999 . . 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 1906 . . 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 1906 . . 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 5155 . . . . . . . 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 8000 . . . . . . 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 5742 . . . . . 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 2974 . . . . . 6 𝑔{𝑐}
5250, 51nfima 5930 . . . . 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 4877 . . . 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 2974 . . . . . 6 𝑔{𝑏}
5550, 54nfima 5930 . . . . 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 4877 . . . 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 2989 . . 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 2974 . . . . . . . . 9 V
59 nfcv 2974 . . . . . . . . . . . 12 (𝑦𝑔)
60 nfcv 2974 . . . . . . . . . . . 12 (𝑅1‘dom 𝑒)
61 nfopab2 5127 . . . . . . . . . . . 12 {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}
6259, 60, 61nfsup 8903 . . . . . . . . . . 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 5154 . . . . . . . . . 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 2974 . . . . . . . . . 10 ((𝑅1‘dom 𝑒) ∖ ran 𝑔)
6563, 64nffv 6673 . . . . . . . . 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 5154 . . . . . . . 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 8000 . . . . . . 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 5742 . . . . . 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 2974 . . . . . 6 {𝑐}
7068, 69nfima 5930 . . . . 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 4877 . . . 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 2974 . . . . . 6 {𝑏}
7368, 72nfima 5930 . . . . 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 4877 . . . 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 2989 . . 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 4567 . . . . . 6 (𝑔 = 𝑐 → {𝑔} = {𝑐})
7776imaeq2d 5922 . . . . 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 4872 . . . 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 4567 . . . . . 6 ( = 𝑏 → {} = {𝑏})
8079imaeq2d 5922 . . . . 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 4872 . . . 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 2899 . . . 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 5128 . 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 6663 . . . . 5 (𝑔 = 𝑐 → (rank‘𝑔) = (rank‘𝑐))
86 fveq2 6663 . . . . 5 ( = 𝑏 → (rank‘) = (rank‘𝑏))
8785, 86breqan12d 5073 . . . 4 ((𝑔 = 𝑐 = 𝑏) → ((rank‘𝑔) E (rank‘) ↔ (rank‘𝑐) E (rank‘𝑏)))
8885, 86eqeqan12d 2835 . . . . 5 ((𝑔 = 𝑐 = 𝑏) → ((rank‘𝑔) = (rank‘) ↔ (rank‘𝑐) = (rank‘𝑏)))
89 simpl 483 . . . . . 6 ((𝑔 = 𝑐 = 𝑏) → 𝑔 = 𝑐)
90 suceq 6249 . . . . . . . . 9 ((rank‘𝑔) = (rank‘𝑐) → suc (rank‘𝑔) = suc (rank‘𝑐))
9185, 90syl 17 . . . . . . . 8 (𝑔 = 𝑐 → suc (rank‘𝑔) = suc (rank‘𝑐))
9291adantr 481 . . . . . . 7 ((𝑔 = 𝑐 = 𝑏) → suc (rank‘𝑔) = suc (rank‘𝑐))
9392fveq2d 6667 . . . . . 6 ((𝑔 = 𝑐 = 𝑏) → (𝑒‘suc (rank‘𝑔)) = (𝑒‘suc (rank‘𝑐)))
94 simpr 485 . . . . . 6 ((𝑔 = 𝑐 = 𝑏) → = 𝑏)
9589, 93, 94breq123d 5071 . . . . 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 912 . . 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 5129 . 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 2818 . 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 5765 . . . . . . 7 (𝑙 = 𝑒 → dom 𝑙 = dom 𝑒)
101100unieqd 4840 . . . . . . 7 (𝑙 = 𝑒 dom 𝑙 = dom 𝑒)
102100, 101eqeq12d 2834 . . . . . 6 (𝑙 = 𝑒 → (dom 𝑙 = dom 𝑙 ↔ dom 𝑒 = dom 𝑒))
103 fveq1 6662 . . . . . . . . . 10 (𝑙 = 𝑒 → (𝑙‘suc (rank‘𝑔)) = (𝑒‘suc (rank‘𝑔)))
104103breqd 5068 . . . . . . . . 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 909 . . . . . . 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 5123 . . . . . 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 2819 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑒 → (𝑦𝑔) = (𝑦𝑔))
109100fveq2d 6667 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑒 → (𝑅1‘dom 𝑙) = (𝑅1‘dom 𝑒))
110101fveq2d 6667 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝑒 → (𝑅1 dom 𝑙) = (𝑅1 dom 𝑒))
111 id 22 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑒𝑙 = 𝑒)
112111, 101fveq12d 6670 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 = 𝑒 → (𝑙 dom 𝑙) = (𝑒 dom 𝑒))
113112breqd 5068 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 = 𝑒 → (𝑗(𝑙 dom 𝑙)𝑖𝑗(𝑒 dom 𝑒)𝑖))
114113imbi1d 343 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = 𝑒 → ((𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)) ↔ (𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗))))
115110, 114raleqbidv 3399 . . . . . . . . . . . . . . . . . . 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 3400 . . . . . . . . . . . . . . . . 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 5123 . . . . . . . . . . . . . . . 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 8902 . . . . . . . . . . . . . . 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 5153 . . . . . . . . . . . . . 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 4095 . . . . . . . . . . . . . 14 (𝑙 = 𝑒 → ((𝑅1‘dom 𝑙) ∖ ran 𝑔) = ((𝑅1‘dom 𝑒) ∖ ran 𝑔))
122120, 121fveq12d 6670 . . . . . . . . . . . . 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 5153 . . . . . . . . . . . 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 7999 . . . . . . . . . . . 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 5739 . . . . . . . . . 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 5921 . . . . . . . . 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 4872 . . . . . . . 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 5921 . . . . . . . . 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 4872 . . . . . . . 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 2904 . . . . . . 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 5123 . . . . . 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 4490 . . . . 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 5580 . . . . 5 (𝑙 = 𝑒 → ((𝑅1‘dom 𝑙) × (𝑅1‘dom 𝑙)) = ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒)))
135133, 134ineq12d 4187 . . . 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 5160 . . 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 7999 . . 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 3075 . . . . 5 (𝑎 = 𝑐 → (𝑎 ≠ ∅ ↔ 𝑐 ≠ ∅))
142 fveq2 6663 . . . . . 6 (𝑎 = 𝑐 → (𝑦𝑎) = (𝑦𝑐))
143 pweq 4538 . . . . . . . 8 (𝑎 = 𝑐 → 𝒫 𝑎 = 𝒫 𝑐)
144143ineq1d 4185 . . . . . . 7 (𝑎 = 𝑐 → (𝒫 𝑎 ∩ Fin) = (𝒫 𝑐 ∩ Fin))
145144difeq1d 4095 . . . . . 6 (𝑎 = 𝑐 → ((𝒫 𝑎 ∩ Fin) ∖ {∅}) = ((𝒫 𝑐 ∩ Fin) ∖ {∅}))
146142, 145eleq12d 2904 . . . . 5 (𝑎 = 𝑐 → ((𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅}) ↔ (𝑦𝑐) ∈ ((𝒫 𝑐 ∩ Fin) ∖ {∅})))
147141, 146imbi12d 346 . . . 4 (𝑎 = 𝑐 → ((𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})) ↔ (𝑐 ≠ ∅ → (𝑦𝑐) ∈ ((𝒫 𝑐 ∩ Fin) ∖ {∅}))))
148147cbvralvw 3447 . . 3 (∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})) ↔ ∀𝑐 ∈ 𝒫 (𝑅1𝐴)(𝑐 ≠ ∅ → (𝑦𝑐) ∈ ((𝒫 𝑐 ∩ Fin) ∖ {∅})))
149140, 148sylib 219 . 2 (𝜑 → ∀𝑐 ∈ 𝒫 (𝑅1𝐴)(𝑐 ≠ ∅ → (𝑦𝑐) ∈ ((𝒫 𝑐 ∩ Fin) ∖ {∅})))
15029, 37, 45, 84, 98, 99, 138, 139, 149aomclem7 39538 1 (𝜑 → ∃𝑏 𝑏 We (𝑅1𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wex 1771  wcel 2105  wne 3013  wral 3135  wrex 3136  Vcvv 3492  cdif 3930  cin 3932  c0 4288  ifcif 4463  𝒫 cpw 4535  {csn 4557   cuni 4830   cint 4867   class class class wbr 5057  {copab 5119  cmpt 5137   E cep 5457   We wwe 5506   × cxp 5546  ccnv 5547  dom cdm 5548  ran crn 5549  cima 5551  Oncon0 6184  suc csuc 6186  cfv 6348  recscrecs 7996  Fincfn 8497  supcsup 8892  𝑅1cr1 9179  rankcrnk 9180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-er 8278  df-map 8397  df-en 8498  df-fin 8501  df-sup 8894  df-r1 9181  df-rank 9182
This theorem is referenced by:  dfac11  39540
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