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Theorem aomclem6 43048
Description: Lemma for dfac11 43051. 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 4018 . 2 𝐴𝐴
2 aomclem6.a . . . 4 (𝜑𝐴 ∈ On)
32adantr 480 . . 3 ((𝜑𝐴𝐴) → 𝐴 ∈ On)
4 sseq1 4021 . . . . . 6 (𝑐 = 𝑑 → (𝑐𝐴𝑑𝐴))
54anbi2d 630 . . . . 5 (𝑐 = 𝑑 → ((𝜑𝑐𝐴) ↔ (𝜑𝑑𝐴)))
6 fveq2 6907 . . . . . 6 (𝑐 = 𝑑 → (𝐻𝑐) = (𝐻𝑑))
7 fveq2 6907 . . . . . 6 (𝑐 = 𝑑 → (𝑅1𝑐) = (𝑅1𝑑))
86, 7weeq12d 5678 . . . . 5 (𝑐 = 𝑑 → ((𝐻𝑐) We (𝑅1𝑐) ↔ (𝐻𝑑) We (𝑅1𝑑)))
95, 8imbi12d 344 . . . 4 (𝑐 = 𝑑 → (((𝜑𝑐𝐴) → (𝐻𝑐) We (𝑅1𝑐)) ↔ ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑))))
10 sseq1 4021 . . . . . 6 (𝑐 = 𝐴 → (𝑐𝐴𝐴𝐴))
1110anbi2d 630 . . . . 5 (𝑐 = 𝐴 → ((𝜑𝑐𝐴) ↔ (𝜑𝐴𝐴)))
12 fveq2 6907 . . . . . 6 (𝑐 = 𝐴 → (𝐻𝑐) = (𝐻𝐴))
13 fveq2 6907 . . . . . 6 (𝑐 = 𝐴 → (𝑅1𝑐) = (𝑅1𝐴))
1412, 13weeq12d 5678 . . . . 5 (𝑐 = 𝐴 → ((𝐻𝑐) We (𝑅1𝑐) ↔ (𝐻𝐴) We (𝑅1𝐴)))
1511, 14imbi12d 344 . . . 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 5917 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝐻𝑐) → dom 𝑧 = dom (𝐻𝑐))
2322adantl 481 . . . . . . . . . . . . . . . 16 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → dom 𝑧 = dom (𝐻𝑐))
24 simpl1 1190 . . . . . . . . . . . . . . . . 17 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → 𝑐 ∈ On)
25 onss 7804 . . . . . . . . . . . . . . . . 17 (𝑐 ∈ On → 𝑐 ⊆ On)
26 aomclem6.h . . . . . . . . . . . . . . . . . . 19 𝐻 = recs((𝑧 ∈ V ↦ 𝐺))
2726tfr1 8436 . . . . . . . . . . . . . . . . . 18 𝐻 Fn On
28 fnssres 6692 . . . . . . . . . . . . . . . . . 18 ((𝐻 Fn On ∧ 𝑐 ⊆ On) → (𝐻𝑐) Fn 𝑐)
2927, 28mpan 690 . . . . . . . . . . . . . . . . 17 (𝑐 ⊆ On → (𝐻𝑐) Fn 𝑐)
30 fndm 6672 . . . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . . 15 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → dom 𝑧 = 𝑐)
3332, 24eqeltrd 2839 . . . . . . . . . . . . . 14 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → dom 𝑧 ∈ On)
3432eleq2d 2825 . . . . . . . . . . . . . . . . . 18 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → (𝑎 ∈ dom 𝑧𝑎𝑐))
3534biimpa 476 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → 𝑎𝑐)
36 simpll2 1212 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)))
37 simpl3l 1227 . . . . . . . . . . . . . . . . . 18 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → 𝜑)
3837adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → 𝜑)
39 onelss 6428 . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . 18 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → 𝑎 ⊆ dom 𝑧)
42 simpl3r 1228 . . . . . . . . . . . . . . . . . . . 20 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → 𝑐𝐴)
4332, 42eqsstrd 4034 . . . . . . . . . . . . . . . . . . 19 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → dom 𝑧𝐴)
4443adantr 480 . . . . . . . . . . . . . . . . . 18 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → dom 𝑧𝐴)
4541, 44sstrd 4006 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → 𝑎𝐴)
46 sseq1 4021 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 = 𝑎 → (𝑑𝐴𝑎𝐴))
4746anbi2d 630 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = 𝑎 → ((𝜑𝑑𝐴) ↔ (𝜑𝑎𝐴)))
48 fveq2 6907 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 = 𝑎 → (𝐻𝑑) = (𝐻𝑎))
49 fveq2 6907 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 = 𝑎 → (𝑅1𝑑) = (𝑅1𝑎))
5048, 49weeq12d 5678 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = 𝑎 → ((𝐻𝑑) We (𝑅1𝑑) ↔ (𝐻𝑎) We (𝑅1𝑎)))
5147, 50imbi12d 344 . . . . . . . . . . . . . . . . . . 19 (𝑑 = 𝑎 → (((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ↔ ((𝜑𝑎𝐴) → (𝐻𝑎) We (𝑅1𝑎))))
5251rspcva 3620 . . . . . . . . . . . . . . . . . 18 ((𝑎𝑐 ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑))) → ((𝜑𝑎𝐴) → (𝐻𝑎) We (𝑅1𝑎)))
5352imp 406 . . . . . . . . . . . . . . . . 17 (((𝑎𝑐 ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑))) ∧ (𝜑𝑎𝐴)) → (𝐻𝑎) We (𝑅1𝑎))
5435, 36, 38, 45, 53syl22anc 839 . . . . . . . . . . . . . . . 16 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → (𝐻𝑎) We (𝑅1𝑎))
55 fveq1 6906 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝐻𝑐) → (𝑧𝑎) = ((𝐻𝑐)‘𝑎))
5655ad2antlr 727 . . . . . . . . . . . . . . . . . 18 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → (𝑧𝑎) = ((𝐻𝑐)‘𝑎))
57 fvres 6926 . . . . . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → (𝑧𝑎) = (𝐻𝑎))
60 weeq1 5676 . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . 15 ((((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) ∧ 𝑎 ∈ dom 𝑧) → (𝑧𝑎) We (𝑅1𝑎))
6362ralrimiva 3144 . . . . . . . . . . . . . 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 43047 . . . . . . . . . . . . 13 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → 𝐺 We (𝑅1‘dom 𝑧))
6832fveq2d 6911 . . . . . . . . . . . . . 14 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → (𝑅1‘dom 𝑧) = (𝑅1𝑐))
69 weeq2 5677 . . . . . . . . . . . . . 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 232 . . . . . . . . . . . 12 (((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) ∧ 𝑧 = (𝐻𝑐)) → 𝐺 We (𝑅1𝑐))
7271ex 412 . . . . . . . . . . 11 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → (𝑧 = (𝐻𝑐) → 𝐺 We (𝑅1𝑐)))
7372alrimiv 1925 . . . . . . . . . 10 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → ∀𝑧(𝑧 = (𝐻𝑐) → 𝐺 We (𝑅1𝑐)))
74 nfv 1912 . . . . . . . . . . 11 𝑑(𝑧 = (𝐻𝑐) → 𝐺 We (𝑅1𝑐))
75 nfv 1912 . . . . . . . . . . . 12 𝑧 𝑑 = (𝐻𝑐)
76 nfsbc1v 3811 . . . . . . . . . . . 12 𝑧[𝑑 / 𝑧]𝐺 We (𝑅1𝑐)
7775, 76nfim 1894 . . . . . . . . . . 11 𝑧(𝑑 = (𝐻𝑐) → [𝑑 / 𝑧]𝐺 We (𝑅1𝑐))
78 eqeq1 2739 . . . . . . . . . . . 12 (𝑧 = 𝑑 → (𝑧 = (𝐻𝑐) ↔ 𝑑 = (𝐻𝑐)))
79 sbceq1a 3802 . . . . . . . . . . . 12 (𝑧 = 𝑑 → (𝐺 We (𝑅1𝑐) ↔ [𝑑 / 𝑧]𝐺 We (𝑅1𝑐)))
8078, 79imbi12d 344 . . . . . . . . . . 11 (𝑧 = 𝑑 → ((𝑧 = (𝐻𝑐) → 𝐺 We (𝑅1𝑐)) ↔ (𝑑 = (𝐻𝑐) → [𝑑 / 𝑧]𝐺 We (𝑅1𝑐))))
8174, 77, 80cbvalv1 2342 . . . . . . . . . 10 (∀𝑧(𝑧 = (𝐻𝑐) → 𝐺 We (𝑅1𝑐)) ↔ ∀𝑑(𝑑 = (𝐻𝑐) → [𝑑 / 𝑧]𝐺 We (𝑅1𝑐)))
8273, 81sylib 218 . . . . . . . . 9 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → ∀𝑑(𝑑 = (𝐻𝑐) → [𝑑 / 𝑧]𝐺 We (𝑅1𝑐)))
83 nfsbc1v 3811 . . . . . . . . . 10 𝑑[(𝐻𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1𝑐)
84 fnfun 6669 . . . . . . . . . . . 12 (𝐻 Fn On → Fun 𝐻)
8527, 84ax-mp 5 . . . . . . . . . . 11 Fun 𝐻
86 vex 3482 . . . . . . . . . . 11 𝑐 ∈ V
87 resfunexg 7235 . . . . . . . . . . 11 ((Fun 𝐻𝑐 ∈ V) → (𝐻𝑐) ∈ V)
8885, 86, 87mp2an 692 . . . . . . . . . 10 (𝐻𝑐) ∈ V
89 sbceq1a 3802 . . . . . . . . . 10 (𝑑 = (𝐻𝑐) → ([𝑑 / 𝑧]𝐺 We (𝑅1𝑐) ↔ [(𝐻𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1𝑐)))
9083, 88, 89ceqsal 3517 . . . . . . . . 9 (∀𝑑(𝑑 = (𝐻𝑐) → [𝑑 / 𝑧]𝐺 We (𝑅1𝑐)) ↔ [(𝐻𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1𝑐))
9182, 90sylib 218 . . . . . . . 8 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → [(𝐻𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1𝑐))
92 sbccow 3814 . . . . . . . 8 ([(𝐻𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1𝑐) ↔ [(𝐻𝑐) / 𝑧]𝐺 We (𝑅1𝑐))
9391, 92sylib 218 . . . . . . 7 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → [(𝐻𝑐) / 𝑧]𝐺 We (𝑅1𝑐))
94 nfcsb1v 3933 . . . . . . . . . 10 𝑧(𝐻𝑐) / 𝑧𝐺
95 nfcv 2903 . . . . . . . . . 10 𝑧(𝑅1𝑐)
9694, 95nfwe 5664 . . . . . . . . 9 𝑧(𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐)
97 csbeq1a 3922 . . . . . . . . . 10 (𝑧 = (𝐻𝑐) → 𝐺 = (𝐻𝑐) / 𝑧𝐺)
98 weeq1 5676 . . . . . . . . . 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 3831 . . . . . . . 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 218 . . . . . 6 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → (𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐))
103 recsval 8443 . . . . . . . . 9 (𝑐 ∈ On → (recs((𝑧 ∈ V ↦ 𝐺))‘𝑐) = ((𝑧 ∈ V ↦ 𝐺)‘(recs((𝑧 ∈ V ↦ 𝐺)) ↾ 𝑐)))
10426fveq1i 6908 . . . . . . . . 9 (𝐻𝑐) = (recs((𝑧 ∈ V ↦ 𝐺))‘𝑐)
105 fvex 6920 . . . . . . . . . . . . . . 15 (𝑅1‘dom 𝑧) ∈ V
106105, 105xpex 7772 . . . . . . . . . . . . . 14 ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)) ∈ V
107106inex2 5324 . . . . . . . . . . . . 13 (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) ∈ V
10821, 107eqeltri 2835 . . . . . . . . . . . 12 𝐺 ∈ V
109108csbex 5317 . . . . . . . . . . 11 (𝐻𝑐) / 𝑧𝐺 ∈ V
110 eqid 2735 . . . . . . . . . . . 12 (𝑧 ∈ V ↦ 𝐺) = (𝑧 ∈ V ↦ 𝐺)
111110fvmpts 7019 . . . . . . . . . . 11 (((𝐻𝑐) ∈ V ∧ (𝐻𝑐) / 𝑧𝐺 ∈ V) → ((𝑧 ∈ V ↦ 𝐺)‘(𝐻𝑐)) = (𝐻𝑐) / 𝑧𝐺)
11288, 109, 111mp2an 692 . . . . . . . . . 10 ((𝑧 ∈ V ↦ 𝐺)‘(𝐻𝑐)) = (𝐻𝑐) / 𝑧𝐺
11326reseq1i 5996 . . . . . . . . . . 11 (𝐻𝑐) = (recs((𝑧 ∈ V ↦ 𝐺)) ↾ 𝑐)
114113fveq2i 6910 . . . . . . . . . 10 ((𝑧 ∈ V ↦ 𝐺)‘(𝐻𝑐)) = ((𝑧 ∈ V ↦ 𝐺)‘(recs((𝑧 ∈ V ↦ 𝐺)) ↾ 𝑐))
115112, 114eqtr3i 2765 . . . . . . . . 9 (𝐻𝑐) / 𝑧𝐺 = ((𝑧 ∈ V ↦ 𝐺)‘(recs((𝑧 ∈ V ↦ 𝐺)) ↾ 𝑐))
116103, 104, 1153eqtr4g 2800 . . . . . . . 8 (𝑐 ∈ On → (𝐻𝑐) = (𝐻𝑐) / 𝑧𝐺)
117 weeq1 5676 . . . . . . . 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 1132 . . . . . 6 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → ((𝐻𝑐) We (𝑅1𝑐) ↔ (𝐻𝑐) / 𝑧𝐺 We (𝑅1𝑐)))
120102, 119mpbird 257 . . . . 5 ((𝑐 ∈ On ∧ ∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) ∧ (𝜑𝑐𝐴)) → (𝐻𝑐) We (𝑅1𝑐))
1211203exp 1118 . . . 4 (𝑐 ∈ On → (∀𝑑𝑐 ((𝜑𝑑𝐴) → (𝐻𝑑) We (𝑅1𝑑)) → ((𝜑𝑐𝐴) → (𝐻𝑐) We (𝑅1𝑐))))
1229, 15, 121tfis3 7879 . . 3 (𝐴 ∈ On → ((𝜑𝐴𝐴) → (𝐻𝐴) We (𝑅1𝐴)))
1233, 122mpcom 38 . 2 ((𝜑𝐴𝐴) → (𝐻𝐴) We (𝑅1𝐴))
1241, 123mpan2 691 1 (𝜑 → (𝐻𝐴) We (𝑅1𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3a 1086  wal 1535   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  [wsbc 3791  csb 3908  cdif 3960  cin 3962  wss 3963  c0 4339  ifcif 4531  𝒫 cpw 4605  {csn 4631   cuni 4912   cint 4951   class class class wbr 5148  {copab 5210  cmpt 5231   E cep 5588   We wwe 5640   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  Oncon0 6386  suc csuc 6388  Fun wfun 6557   Fn wfn 6558  cfv 6563  recscrecs 8409  Fincfn 8984  supcsup 9478  𝑅1cr1 9800  rankcrnk 9801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-map 8867  df-en 8985  df-fin 8988  df-sup 9480  df-r1 9802  df-rank 9803
This theorem is referenced by:  aomclem7  43049
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