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Theorem aomclem6 38127
Description: Lemma for dfac11 38130. Transfinite induction, close over 𝑧. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Hypotheses
Ref Expression
aomclem6.b 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1 dom 𝑧)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}
aomclem6.c 𝐶 = (𝑎 ∈ V ↦ sup((𝑦𝑎), (𝑅1‘dom 𝑧), 𝐵))
aomclem6.d 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))
aomclem6.e 𝐸 = {⟨𝑎, 𝑏⟩ ∣ (𝐷 “ {𝑎}) ∈ (𝐷 “ {𝑏})}
aomclem6.f 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}
aomclem6.g 𝐺 = (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)))
aomclem6.h 𝐻 = recs((𝑧 ∈ V ↦ 𝐺))
aomclem6.a (𝜑𝐴 ∈ On)
aomclem6.y (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
Assertion
Ref Expression
aomclem6 (𝜑 → (𝐻𝐴) We (𝑅1𝐴))
Distinct variable groups:   𝑦,𝑧,𝑎,𝑏,𝑐,𝑑   𝜑,𝑎,𝑏,𝑐,𝑑,𝑧   𝐶,𝑎,𝑏,𝑐,𝑑   𝐷,𝑎,𝑏,𝑐,𝑑   𝐴,𝑎,𝑏,𝑐,𝑑,𝑧   𝐻,𝑎,𝑏,𝑐,𝑑,𝑧   𝐺,𝑑
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)   𝐶(𝑦,𝑧)   𝐷(𝑦,𝑧)   𝐸(𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)   𝐹(𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)   𝐺(𝑦,𝑧,𝑎,𝑏,𝑐)   𝐻(𝑦)

Proof of Theorem aomclem6
StepHypRef Expression
1 ssid 3761 . 2 𝐴𝐴
2 aomclem6.a . . . 4 (𝜑𝐴 ∈ On)
32adantr 472 . . 3 ((𝜑𝐴𝐴) → 𝐴 ∈ On)
4 sseq1 3763 . . . . . 6 (𝑐 = 𝑑 → (𝑐𝐴𝑑𝐴))
54anbi2d 742 . . . . 5 (𝑐 = 𝑑 → ((𝜑𝑐𝐴) ↔ (𝜑𝑑𝐴)))
6 fveq2 6348 . . . . . 6 (𝑐 = 𝑑 → (𝐻𝑐) = (𝐻𝑑))
7 fveq2 6348 . . . . . 6 (𝑐 = 𝑑 → (𝑅1𝑐) = (𝑅1𝑑))
86, 7weeq12d 38108 . . . . 5 (𝑐 = 𝑑 → ((𝐻𝑐) We (𝑅1𝑐) ↔ (𝐻𝑑) We (𝑅1𝑑)))
95, 8imbi12d 333 . . . 4 (𝑐 = 𝑑 → (((𝜑𝑐𝐴) → (𝐻𝑐) We (𝑅1𝑐)) ↔ ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑))))
10 sseq1 3763 . . . . . 6 (𝑐 = 𝐴 → (𝑐𝐴𝐴𝐴))
1110anbi2d 742 . . . . 5 (𝑐 = 𝐴 → ((𝜑𝑐𝐴) ↔ (𝜑𝐴𝐴)))
12 fveq2 6348 . . . . . 6 (𝑐 = 𝐴 → (𝐻𝑐) = (𝐻𝐴))
13 fveq2 6348 . . . . . 6 (𝑐 = 𝐴 → (𝑅1𝑐) = (𝑅1𝐴))
1412, 13weeq12d 38108 . . . . 5 (𝑐 = 𝐴 → ((𝐻𝑐) We (𝑅1𝑐) ↔ (𝐻𝐴) We (𝑅1𝐴)))
1511, 14imbi12d 333 . . . 4 (𝑐 = 𝐴 → (((𝜑𝑐𝐴) → (𝐻𝑐) We (𝑅1𝑐)) ↔ ((𝜑𝐴𝐴) → (𝐻𝐴) We (𝑅1𝐴))))
16 aomclem6.b . . . . . . . . . . . . . 14 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1 dom 𝑧)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}
17 aomclem6.c . . . . . . . . . . . . . 14 𝐶 = (𝑎 ∈ V ↦ sup((𝑦𝑎), (𝑅1‘dom 𝑧), 𝐵))
18 aomclem6.d . . . . . . . . . . . . . 14 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))
19 aomclem6.e . . . . . . . . . . . . . 14 𝐸 = {⟨𝑎, 𝑏⟩ ∣ (𝐷 “ {𝑎}) ∈ (𝐷 “ {𝑏})}
20 aomclem6.f . . . . . . . . . . . . . 14 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}
21 aomclem6.g . . . . . . . . . . . . . 14 𝐺 = (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)))
22 dmeq 5475 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝐻𝑐) → dom 𝑧 = dom (𝐻𝑐))
2322adantl 473 . . . . . . . . . . . . . . . 16 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → dom 𝑧 = dom (𝐻𝑐))
24 simpl1 1228 . . . . . . . . . . . . . . . . 17 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → 𝑐 ∈ On)
25 onss 7151 . . . . . . . . . . . . . . . . 17 (𝑐 ∈ On → 𝑐 ⊆ On)
26 aomclem6.h . . . . . . . . . . . . . . . . . . 19 𝐻 = recs((𝑧 ∈ V ↦ 𝐺))
2726tfr1 7658 . . . . . . . . . . . . . . . . . 18 𝐻 Fn On
28 fnssres 6161 . . . . . . . . . . . . . . . . . 18 ((𝐻 Fn On ∧ 𝑐 ⊆ On) → (𝐻𝑐) Fn 𝑐)
2927, 28mpan 708 . . . . . . . . . . . . . . . . 17 (𝑐 ⊆ On → (𝐻𝑐) Fn 𝑐)
30 fndm 6147 . . . . . . . . . . . . . . . . 17 ((𝐻𝑐) Fn 𝑐 → dom (𝐻𝑐) = 𝑐)
3124, 25, 29, 304syl 19 . . . . . . . . . . . . . . . 16 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → dom (𝐻𝑐) = 𝑐)
3223, 31eqtrd 2790 . . . . . . . . . . . . . . 15 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → dom 𝑧 = 𝑐)
3332, 24eqeltrd 2835 . . . . . . . . . . . . . 14 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → dom 𝑧 ∈ On)
3432eleq2d 2821 . . . . . . . . . . . . . . . . . 18 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → (𝑎 ∈ dom 𝑧𝑎𝑐))
3534biimpa 502 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → 𝑎𝑐)
36 simpll2 1257 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)))
37 simpl3l 1287 . . . . . . . . . . . . . . . . . 18 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → 𝜑)
3837adantr 472 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → 𝜑)
39 onelss 5923 . . . . . . . . . . . . . . . . . . . 20 (dom 𝑧 ∈ On → (𝑎 ∈ dom 𝑧𝑎 ⊆ dom 𝑧))
4033, 39syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → (𝑎 ∈ dom 𝑧𝑎 ⊆ dom 𝑧))
4140imp 444 . . . . . . . . . . . . . . . . . 18 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → 𝑎 ⊆ dom 𝑧)
42 simpl3r 1289 . . . . . . . . . . . . . . . . . . . 20 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → 𝑐𝐴)
4332, 42eqsstrd 3776 . . . . . . . . . . . . . . . . . . 19 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → dom 𝑧𝐴)
4443adantr 472 . . . . . . . . . . . . . . . . . 18 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → dom 𝑧𝐴)
4541, 44sstrd 3750 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → 𝑎𝐴)
46 sseq1 3763 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 = 𝑎 → (𝑑𝐴𝑎𝐴))
4746anbi2d 742 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = 𝑎 → ((𝜑𝑑𝐴) ↔ (𝜑𝑎𝐴)))
48 fveq2 6348 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 = 𝑎 → (𝐻𝑑) = (𝐻𝑎))
49 fveq2 6348 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 = 𝑎 → (𝑅1𝑑) = (𝑅1𝑎))
5048, 49weeq12d 38108 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = 𝑎 → ((𝐻𝑑) We (𝑅1𝑑) ↔ (𝐻𝑎) We (𝑅1𝑎)))
5147, 50imbi12d 333 . . . . . . . . . . . . . . . . . . 19 (𝑑 = 𝑎 → (((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ↔ ((𝜑𝑎𝐴) → (𝐻𝑎) We (𝑅1𝑎))))
5251rspcva 3443 . . . . . . . . . . . . . . . . . 18 ((𝑎𝑐 ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑))) → ((𝜑𝑎𝐴) → (𝐻𝑎) We (𝑅1𝑎)))
5352imp 444 . . . . . . . . . . . . . . . . 17 (((𝑎𝑐 ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑))) ∧ (𝜑𝑎𝐴)) → (𝐻𝑎) We (𝑅1𝑎))
5435, 36, 38, 45, 53syl22anc 1478 . . . . . . . . . . . . . . . 16 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → (𝐻𝑎) We (𝑅1𝑎))
55 fveq1 6347 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝐻𝑐) → (𝑧𝑎) = ((𝐻𝑐)‘𝑎))
5655ad2antlr 765 . . . . . . . . . . . . . . . . . 18 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → (𝑧𝑎) = ((𝐻𝑐)‘𝑎))
57 fvres 6364 . . . . . . . . . . . . . . . . . . 19 (𝑎𝑐 → ((𝐻𝑐)‘𝑎) = (𝐻𝑎))
5835, 57syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → ((𝐻𝑐)‘𝑎) = (𝐻𝑎))
5956, 58eqtrd 2790 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → (𝑧𝑎) = (𝐻𝑎))
60 weeq1 5250 . . . . . . . . . . . . . . . . 17 ((𝑧𝑎) = (𝐻𝑎) → ((𝑧𝑎) We (𝑅1𝑎) ↔ (𝐻𝑎) We (𝑅1𝑎)))
6159, 60syl 17 . . . . . . . . . . . . . . . 16 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → ((𝑧𝑎) We (𝑅1𝑎) ↔ (𝐻𝑎) We (𝑅1𝑎)))
6254, 61mpbird 247 . . . . . . . . . . . . . . 15 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → (𝑧𝑎) We (𝑅1𝑎))
6362ralrimiva 3100 . . . . . . . . . . . . . 14 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
6437, 2syl 17 . . . . . . . . . . . . . 14 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → 𝐴 ∈ On)
65 aomclem6.y . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
6637, 65syl 17 . . . . . . . . . . . . . 14 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
6716, 17, 18, 19, 20, 21, 33, 63, 64, 43, 66aomclem5 38126 . . . . . . . . . . . . 13 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → 𝐺 We (𝑅1‘dom 𝑧))
6832fveq2d 6352 . . . . . . . . . . . . . 14 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → (𝑅1‘dom 𝑧) = (𝑅1𝑐))
69 weeq2 5251 . . . . . . . . . . . . . 14 ((𝑅1‘dom 𝑧) = (𝑅1𝑐) → (𝐺 We (𝑅1‘dom 𝑧) ↔ 𝐺 We (𝑅1𝑐)))
7068, 69syl 17 . . . . . . . . . . . . 13 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → (𝐺 We (𝑅1‘dom 𝑧) ↔ 𝐺 We (𝑅1𝑐)))
7167, 70mpbid 222 . . . . . . . . . . . 12 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → 𝐺 We (𝑅1𝑐))
7271ex 449 . . . . . . . . . . 11 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → (𝑧 = (𝐻𝑐) → 𝐺 We (𝑅1𝑐)))
7372alrimiv 2000 . . . . . . . . . 10 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → ∀𝑧(𝑧 = (𝐻𝑐) → 𝐺 We (𝑅1𝑐)))
74 nfv 1988 . . . . . . . . . . 11 𝑑(𝑧 = (𝐻𝑐) → 𝐺 We (𝑅1𝑐))
75 nfv 1988 . . . . . . . . . . . 12 𝑧 𝑑 = (𝐻𝑐)
76 nfsbc1v 3592 . . . . . . . . . . . 12 𝑧[𝑑 / 𝑧]𝐺 We (𝑅1𝑐)
7775, 76nfim 1970 . . . . . . . . . . 11 𝑧(𝑑 = (𝐻𝑐) → [𝑑 / 𝑧]𝐺 We (𝑅1𝑐))
78 eqeq1 2760 . . . . . . . . . . . 12 (𝑧 = 𝑑 → (𝑧 = (𝐻𝑐) ↔ 𝑑 = (𝐻𝑐)))
79 sbceq1a 3583 . . . . . . . . . . . 12 (𝑧 = 𝑑 → (𝐺 We (𝑅1𝑐) ↔ [𝑑 / 𝑧]𝐺 We (𝑅1𝑐)))
8078, 79imbi12d 333 . . . . . . . . . . 11 (𝑧 = 𝑑 → ((𝑧 = (𝐻𝑐) → 𝐺 We (𝑅1𝑐)) ↔ (𝑑 = (𝐻𝑐) → [𝑑 / 𝑧]𝐺 We (𝑅1𝑐))))
8174, 77, 80cbval 2412 . . . . . . . . . 10 (∀𝑧(𝑧 = (𝐻𝑐) → 𝐺 We (𝑅1𝑐)) ↔ ∀𝑑(𝑑 = (𝐻𝑐) → [𝑑 / 𝑧]𝐺 We (𝑅1𝑐)))
8273, 81sylib 208 . . . . . . . . 9 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → ∀𝑑(𝑑 = (𝐻𝑐) → [𝑑 / 𝑧]𝐺 We (𝑅1𝑐)))
83 nfsbc1v 3592 . . . . . . . . . 10 𝑑[(𝐻𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1𝑐)
84 fnfun 6145 . . . . . . . . . . . 12 (𝐻 Fn On → Fun 𝐻)
8527, 84ax-mp 5 . . . . . . . . . . 11 Fun 𝐻
86 vex 3339 . . . . . . . . . . 11 𝑐 ∈ V
87 resfunexg 6639 . . . . . . . . . . 11 ((Fun 𝐻𝑐 ∈ V) → (𝐻𝑐) ∈ V)
8885, 86, 87mp2an 710 . . . . . . . . . 10 (𝐻𝑐) ∈ V
89 sbceq1a 3583 . . . . . . . . . 10 (𝑑 = (𝐻𝑐) → ([𝑑 / 𝑧]𝐺 We (𝑅1𝑐) ↔ [(𝐻𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1𝑐)))
9083, 88, 89ceqsal 3368 . . . . . . . . 9 (∀𝑑(𝑑 = (𝐻𝑐) → [𝑑 / 𝑧]𝐺 We (𝑅1𝑐)) ↔ [(𝐻𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1𝑐))
9182, 90sylib 208 . . . . . . . 8 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → [(𝐻𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1𝑐))
92 sbcco 3595 . . . . . . . 8 ([(𝐻𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1𝑐) ↔ [(𝐻𝑐) / 𝑧]𝐺 We (𝑅1𝑐))
9391, 92sylib 208 . . . . . . 7 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → [(𝐻𝑐) / 𝑧]𝐺 We (𝑅1𝑐))
94 nfcsb1v 3686 . . . . . . . . . 10 𝑧(𝐻𝑐) / 𝑧𝐺
95 nfcv 2898 . . . . . . . . . 10 𝑧(𝑅1𝑐)
9694, 95nfwe 5238 . . . . . . . . 9 𝑧(𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐)
97 csbeq1a 3679 . . . . . . . . . 10 (𝑧 = (𝐻𝑐) → 𝐺 = (𝐻𝑐) / 𝑧𝐺)
98 weeq1 5250 . . . . . . . . . 10 (𝐺 = (𝐻𝑐) / 𝑧𝐺 → (𝐺 We (𝑅1𝑐) ↔ (𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐)))
9997, 98syl 17 . . . . . . . . 9 (𝑧 = (𝐻𝑐) → (𝐺 We (𝑅1𝑐) ↔ (𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐)))
10096, 99sbciegf 3604 . . . . . . . 8 ((𝐻𝑐) ∈ V → ([(𝐻𝑐) / 𝑧]𝐺 We (𝑅1𝑐) ↔ (𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐)))
10188, 100ax-mp 5 . . . . . . 7 ([(𝐻𝑐) / 𝑧]𝐺 We (𝑅1𝑐) ↔ (𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐))
10293, 101sylib 208 . . . . . 6 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → (𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐))
103 recsval 7665 . . . . . . . . 9 (𝑐 ∈ On → (recs((𝑧 ∈ V ↦ 𝐺))‘𝑐) = ((𝑧 ∈ V ↦ 𝐺)‘(recs((𝑧 ∈ V ↦ 𝐺)) ↾ 𝑐)))
10426fveq1i 6349 . . . . . . . . 9 (𝐻𝑐) = (recs((𝑧 ∈ V ↦ 𝐺))‘𝑐)
105 fvex 6358 . . . . . . . . . . . . . . 15 (𝑅1‘dom 𝑧) ∈ V
106105, 105xpex 7123 . . . . . . . . . . . . . 14 ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)) ∈ V
107106inex2 4948 . . . . . . . . . . . . 13 (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) ∈ V
10821, 107eqeltri 2831 . . . . . . . . . . . 12 𝐺 ∈ V
109108csbex 4941 . . . . . . . . . . 11 (𝐻𝑐) / 𝑧𝐺 ∈ V
110 eqid 2756 . . . . . . . . . . . 12 (𝑧 ∈ V ↦ 𝐺) = (𝑧 ∈ V ↦ 𝐺)
111110fvmpts 6443 . . . . . . . . . . 11 (((𝐻𝑐) ∈ V ∧ (𝐻𝑐) / 𝑧𝐺 ∈ V) → ((𝑧 ∈ V ↦ 𝐺)‘(𝐻𝑐)) = (𝐻𝑐) / 𝑧𝐺)
11288, 109, 111mp2an 710 . . . . . . . . . 10 ((𝑧 ∈ V ↦ 𝐺)‘(𝐻𝑐)) = (𝐻𝑐) / 𝑧𝐺
11326reseq1i 5543 . . . . . . . . . . 11 (𝐻𝑐) = (recs((𝑧 ∈ V ↦ 𝐺)) ↾ 𝑐)
114113fveq2i 6351 . . . . . . . . . 10 ((𝑧 ∈ V ↦ 𝐺)‘(𝐻𝑐)) = ((𝑧 ∈ V ↦ 𝐺)‘(recs((𝑧 ∈ V ↦ 𝐺)) ↾ 𝑐))
115112, 114eqtr3i 2780 . . . . . . . . 9 (𝐻𝑐) / 𝑧𝐺 = ((𝑧 ∈ V ↦ 𝐺)‘(recs((𝑧 ∈ V ↦ 𝐺)) ↾ 𝑐))
116103, 104, 1153eqtr4g 2815 . . . . . . . 8 (𝑐 ∈ On → (𝐻𝑐) = (𝐻𝑐) / 𝑧𝐺)
117 weeq1 5250 . . . . . . . 8 ((𝐻𝑐) = (𝐻𝑐) / 𝑧𝐺 → ((𝐻𝑐) We (𝑅1𝑐) ↔ (𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐)))
118116, 117syl 17 . . . . . . 7 (𝑐 ∈ On → ((𝐻𝑐) We (𝑅1𝑐) ↔ (𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐)))
1191183ad2ant1 1128 . . . . . 6 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → ((𝐻𝑐) We (𝑅1𝑐) ↔ (𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐)))
120102, 119mpbird 247 . . . . 5 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → (𝐻𝑐) We (𝑅1𝑐))
1211203exp 1113 . . . 4 (𝑐 ∈ On → (∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) → ((𝜑𝑐𝐴) → (𝐻𝑐) We (𝑅1𝑐))))
1229, 15, 121tfis3 7218 . . 3 (𝐴 ∈ On → ((𝜑𝐴𝐴) → (𝐻𝐴) We (𝑅1𝐴)))
1233, 122mpcom 38 . 2 ((𝜑𝐴𝐴) → (𝐻𝐴) We (𝑅1𝐴))
1241, 123mpan2 709 1 (𝜑 → (𝐻𝐴) We (𝑅1𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1072  wal 1626   = wceq 1628  wcel 2135  wne 2928  wral 3046  wrex 3047  Vcvv 3336  [wsbc 3572  csb 3670  cdif 3708  cin 3710  wss 3711  c0 4054  ifcif 4226  𝒫 cpw 4298  {csn 4317   cuni 4584   cint 4623   class class class wbr 4800  {copab 4860  cmpt 4877   E cep 5174   We wwe 5220   × cxp 5260  ccnv 5261  dom cdm 5262  ran crn 5263  cres 5264  cima 5265  Oncon0 5880  suc csuc 5882  Fun wfun 6039   Fn wfn 6040  cfv 6045  recscrecs 7632  Fincfn 8117  supcsup 8507  𝑅1cr1 8794  rankcrnk 8795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-fal 1634  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rmo 3054  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-int 4624  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-isom 6054  df-riota 6770  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-om 7227  df-1st 7329  df-2nd 7330  df-wrecs 7572  df-recs 7633  df-rdg 7671  df-1o 7725  df-2o 7726  df-er 7907  df-map 8021  df-en 8118  df-fin 8121  df-sup 8509  df-r1 8796  df-rank 8797
This theorem is referenced by:  aomclem7  38128
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