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Theorem frrlem13 8289
Description: Lemma for well-founded recursion. Assuming that 𝑆 is a subset of 𝐴 and that 𝑧 is 𝑅-minimal, then 𝐶 is an acceptable function. (Contributed by Scott Fenton, 7-Dec-2022.)
Hypotheses
Ref Expression
frrlem11.1 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
frrlem11.2 𝐹 = frecs(𝑅, 𝐴, 𝐺)
frrlem11.3 ((𝜑 ∧ (𝑔𝐵𝐵)) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
frrlem11.4 𝐶 = ((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
frrlem12.5 (𝜑𝑅 Fr 𝐴)
frrlem12.6 ((𝜑𝑧𝐴) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆)
frrlem12.7 ((𝜑𝑧𝐴) → ∀𝑤𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
frrlem13.8 ((𝜑𝑧𝐴) → 𝑆 ∈ V)
frrlem13.9 ((𝜑𝑧𝐴) → 𝑆𝐴)
Assertion
Ref Expression
frrlem13 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶𝐵)
Distinct variable groups:   𝐴,𝑓,𝑥,𝑦,𝑧   𝑓,𝐺,𝑥,𝑦,𝑧   𝑅,𝑓,𝑥,𝑦,𝑧   𝐵,𝑔,,𝑧   𝑥,𝐹,𝑢,𝑣,𝑧   𝜑,𝑓,𝑧   𝑓,𝐹   𝜑,𝑔,,𝑥,𝑢,𝑣   𝐴,,𝑤,𝑓,𝑦,𝑥   𝑤,𝐺   𝑤,𝑅   𝑦,𝐹   𝑥,𝐵   𝑤,𝐶   𝑤,𝐹   𝜑,𝑤   𝑤,𝑆   𝑧,𝑤
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑣,𝑢,𝑔)   𝐵(𝑦,𝑤,𝑣,𝑢,𝑓)   𝐶(𝑥,𝑦,𝑧,𝑣,𝑢,𝑓,𝑔,)   𝑅(𝑣,𝑢,𝑔,)   𝑆(𝑥,𝑦,𝑧,𝑣,𝑢,𝑓,𝑔,)   𝐹(𝑔,)   𝐺(𝑣,𝑢,𝑔,)

Proof of Theorem frrlem13
Dummy variables 𝑐 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifi 4126 . . . . . 6 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝑧𝐴)
2 frrlem13.8 . . . . . 6 ((𝜑𝑧𝐴) → 𝑆 ∈ V)
31, 2sylan2 592 . . . . 5 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝑆 ∈ V)
43adantrr 714 . . . 4 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝑆 ∈ V)
5 inex1g 5319 . . . . 5 (𝑆 ∈ V → (𝑆 ∩ dom 𝐹) ∈ V)
6 snex 5431 . . . . 5 {𝑧} ∈ V
7 unexg 7740 . . . . 5 (((𝑆 ∩ dom 𝐹) ∈ V ∧ {𝑧} ∈ V) → ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ∈ V)
85, 6, 7sylancl 585 . . . 4 (𝑆 ∈ V → ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ∈ V)
94, 8syl 17 . . 3 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ∈ V)
10 frrlem11.1 . . . . 5 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
11 frrlem11.2 . . . . 5 𝐹 = frecs(𝑅, 𝐴, 𝐺)
12 frrlem11.3 . . . . 5 ((𝜑 ∧ (𝑔𝐵𝐵)) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
13 frrlem11.4 . . . . 5 𝐶 = ((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
1410, 11, 12, 13frrlem11 8287 . . . 4 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))
1514adantrr 714 . . 3 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))
16 inss1 4228 . . . . . 6 (𝑆 ∩ dom 𝐹) ⊆ 𝑆
17 frrlem13.9 . . . . . . . 8 ((𝜑𝑧𝐴) → 𝑆𝐴)
181, 17sylan2 592 . . . . . . 7 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝑆𝐴)
1918adantrr 714 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝑆𝐴)
2016, 19sstrid 3993 . . . . 5 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → (𝑆 ∩ dom 𝐹) ⊆ 𝐴)
211adantl 481 . . . . . . 7 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝑧𝐴)
2221adantrr 714 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝑧𝐴)
2322snssd 4812 . . . . 5 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → {𝑧} ⊆ 𝐴)
2420, 23unssd 4186 . . . 4 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ⊆ 𝐴)
25 elun 4148 . . . . . . . . 9 (𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ↔ (𝑤 ∈ (𝑆 ∩ dom 𝐹) ∨ 𝑤 ∈ {𝑧}))
26 elin 3964 . . . . . . . . . 10 (𝑤 ∈ (𝑆 ∩ dom 𝐹) ↔ (𝑤𝑆𝑤 ∈ dom 𝐹))
27 velsn 4644 . . . . . . . . . 10 (𝑤 ∈ {𝑧} ↔ 𝑤 = 𝑧)
2826, 27orbi12i 912 . . . . . . . . 9 ((𝑤 ∈ (𝑆 ∩ dom 𝐹) ∨ 𝑤 ∈ {𝑧}) ↔ ((𝑤𝑆𝑤 ∈ dom 𝐹) ∨ 𝑤 = 𝑧))
2925, 28bitri 275 . . . . . . . 8 (𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ↔ ((𝑤𝑆𝑤 ∈ dom 𝐹) ∨ 𝑤 = 𝑧))
30 frrlem12.7 . . . . . . . . . . . . . 14 ((𝜑𝑧𝐴) → ∀𝑤𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
311, 30sylan2 592 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ∀𝑤𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
3231adantrr 714 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ∀𝑤𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
33 rsp 3243 . . . . . . . . . . . 12 (∀𝑤𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆 → (𝑤𝑆 → Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆))
3432, 33syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → (𝑤𝑆 → Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆))
3510, 11frrlem8 8284 . . . . . . . . . . 11 (𝑤 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹)
3634, 35anim12d1 609 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ((𝑤𝑆𝑤 ∈ dom 𝐹) → (Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆 ∧ Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹)))
37 ssin 4230 . . . . . . . . . 10 ((Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆 ∧ Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹) ↔ Pred(𝑅, 𝐴, 𝑤) ⊆ (𝑆 ∩ dom 𝐹))
3836, 37imbitrdi 250 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ((𝑤𝑆𝑤 ∈ dom 𝐹) → Pred(𝑅, 𝐴, 𝑤) ⊆ (𝑆 ∩ dom 𝐹)))
39 frrlem12.6 . . . . . . . . . . . . 13 ((𝜑𝑧𝐴) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆)
401, 39sylan2 592 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆)
4140adantrr 714 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆)
42 preddif 6330 . . . . . . . . . . . . . . . 16 Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = (Pred(𝑅, 𝐴, 𝑧) ∖ Pred(𝑅, dom 𝐹, 𝑧))
4342eqeq1i 2736 . . . . . . . . . . . . . . 15 (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ ↔ (Pred(𝑅, 𝐴, 𝑧) ∖ Pred(𝑅, dom 𝐹, 𝑧)) = ∅)
44 ssdif0 4363 . . . . . . . . . . . . . . 15 (Pred(𝑅, 𝐴, 𝑧) ⊆ Pred(𝑅, dom 𝐹, 𝑧) ↔ (Pred(𝑅, 𝐴, 𝑧) ∖ Pred(𝑅, dom 𝐹, 𝑧)) = ∅)
4543, 44sylbb2 237 . . . . . . . . . . . . . 14 (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ → Pred(𝑅, 𝐴, 𝑧) ⊆ Pred(𝑅, dom 𝐹, 𝑧))
46 predss 6308 . . . . . . . . . . . . . 14 Pred(𝑅, dom 𝐹, 𝑧) ⊆ dom 𝐹
4745, 46sstrdi 3994 . . . . . . . . . . . . 13 (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
4847adantl 481 . . . . . . . . . . . 12 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
4948adantl 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
5041, 49ssind 4232 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → Pred(𝑅, 𝐴, 𝑧) ⊆ (𝑆 ∩ dom 𝐹))
51 predeq3 6304 . . . . . . . . . . 11 (𝑤 = 𝑧 → Pred(𝑅, 𝐴, 𝑤) = Pred(𝑅, 𝐴, 𝑧))
5251sseq1d 4013 . . . . . . . . . 10 (𝑤 = 𝑧 → (Pred(𝑅, 𝐴, 𝑤) ⊆ (𝑆 ∩ dom 𝐹) ↔ Pred(𝑅, 𝐴, 𝑧) ⊆ (𝑆 ∩ dom 𝐹)))
5350, 52syl5ibrcom 246 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → (𝑤 = 𝑧 → Pred(𝑅, 𝐴, 𝑤) ⊆ (𝑆 ∩ dom 𝐹)))
5438, 53jaod 856 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → (((𝑤𝑆𝑤 ∈ dom 𝐹) ∨ 𝑤 = 𝑧) → Pred(𝑅, 𝐴, 𝑤) ⊆ (𝑆 ∩ dom 𝐹)))
5529, 54biimtrid 241 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → (𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → Pred(𝑅, 𝐴, 𝑤) ⊆ (𝑆 ∩ dom 𝐹)))
5655imp 406 . . . . . 6 (((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) ∧ 𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) → Pred(𝑅, 𝐴, 𝑤) ⊆ (𝑆 ∩ dom 𝐹))
57 ssun1 4172 . . . . . 6 (𝑆 ∩ dom 𝐹) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})
5856, 57sstrdi 3994 . . . . 5 (((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) ∧ 𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) → Pred(𝑅, 𝐴, 𝑤) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))
5958ralrimiva 3145 . . . 4 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})Pred(𝑅, 𝐴, 𝑤) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))
6024, 59jca 511 . . 3 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → (((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})Pred(𝑅, 𝐴, 𝑤) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})))
61 frrlem12.5 . . . . . . 7 (𝜑𝑅 Fr 𝐴)
6210, 11, 12, 13, 61, 39, 30frrlem12 8288 . . . . . 6 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) → (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))
63623expa 1117 . . . . 5 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) → (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))
6463ralrimiva 3145 . . . 4 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})(𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))
6564adantrr 714 . . 3 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})(𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))
66 fneq2 6641 . . . . . 6 (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → (𝐶 Fn 𝑡𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧})))
67 sseq1 4007 . . . . . . 7 (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → (𝑡𝐴 ↔ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ⊆ 𝐴))
68 sseq2 4008 . . . . . . . 8 (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → (Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡 ↔ Pred(𝑅, 𝐴, 𝑤) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})))
6968raleqbi1dv 3332 . . . . . . 7 (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → (∀𝑤𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡 ↔ ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})Pred(𝑅, 𝐴, 𝑤) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})))
7067, 69anbi12d 630 . . . . . 6 (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → ((𝑡𝐴 ∧ ∀𝑤𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ↔ (((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})Pred(𝑅, 𝐴, 𝑤) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))))
71 raleq 3321 . . . . . 6 (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → (∀𝑤𝑡 (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))) ↔ ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})(𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))
7266, 70, 713anbi123d 1435 . . . . 5 (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → ((𝐶 Fn 𝑡 ∧ (𝑡𝐴 ∧ ∀𝑤𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤𝑡 (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))) ↔ (𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ∧ (((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})Pred(𝑅, 𝐴, 𝑤) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) ∧ ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})(𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))))
7372spcegv 3587 . . . 4 (((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ∈ V → ((𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ∧ (((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})Pred(𝑅, 𝐴, 𝑤) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) ∧ ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})(𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))) → ∃𝑡(𝐶 Fn 𝑡 ∧ (𝑡𝐴 ∧ ∀𝑤𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤𝑡 (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))))
7473imp 406 . . 3 ((((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ∈ V ∧ (𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ∧ (((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})Pred(𝑅, 𝐴, 𝑤) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) ∧ ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})(𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))) → ∃𝑡(𝐶 Fn 𝑡 ∧ (𝑡𝐴 ∧ ∀𝑤𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤𝑡 (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))
759, 15, 60, 65, 74syl13anc 1371 . 2 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ∃𝑡(𝐶 Fn 𝑡 ∧ (𝑡𝐴 ∧ ∀𝑤𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤𝑡 (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))
7610, 11, 12frrlem9 8285 . . . . . 6 (𝜑 → Fun 𝐹)
77 resfunexg 7219 . . . . . 6 ((Fun 𝐹𝑆 ∈ V) → (𝐹𝑆) ∈ V)
7876, 4, 77syl2an2r 682 . . . . 5 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → (𝐹𝑆) ∈ V)
79 snex 5431 . . . . 5 {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ∈ V
80 unexg 7740 . . . . 5 (((𝐹𝑆) ∈ V ∧ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ∈ V) → ((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ∈ V)
8178, 79, 80sylancl 585 . . . 4 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ∈ V)
8213, 81eqeltrid 2836 . . 3 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶 ∈ V)
83 fneq1 6640 . . . . . 6 (𝑐 = 𝐶 → (𝑐 Fn 𝑡𝐶 Fn 𝑡))
84 fveq1 6890 . . . . . . . 8 (𝑐 = 𝐶 → (𝑐𝑤) = (𝐶𝑤))
85 reseq1 5975 . . . . . . . . 9 (𝑐 = 𝐶 → (𝑐 ↾ Pred(𝑅, 𝐴, 𝑤)) = (𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))
8685oveq2d 7428 . . . . . . . 8 (𝑐 = 𝐶 → (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤))) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))
8784, 86eqeq12d 2747 . . . . . . 7 (𝑐 = 𝐶 → ((𝑐𝑤) = (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤))) ↔ (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))
8887ralbidv 3176 . . . . . 6 (𝑐 = 𝐶 → (∀𝑤𝑡 (𝑐𝑤) = (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤))) ↔ ∀𝑤𝑡 (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))
8983, 883anbi13d 1437 . . . . 5 (𝑐 = 𝐶 → ((𝑐 Fn 𝑡 ∧ (𝑡𝐴 ∧ ∀𝑤𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤𝑡 (𝑐𝑤) = (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤)))) ↔ (𝐶 Fn 𝑡 ∧ (𝑡𝐴 ∧ ∀𝑤𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤𝑡 (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))))
9089exbidv 1923 . . . 4 (𝑐 = 𝐶 → (∃𝑡(𝑐 Fn 𝑡 ∧ (𝑡𝐴 ∧ ∀𝑤𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤𝑡 (𝑐𝑤) = (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤)))) ↔ ∃𝑡(𝐶 Fn 𝑡 ∧ (𝑡𝐴 ∧ ∀𝑤𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤𝑡 (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))))
9110frrlem1 8277 . . . 4 𝐵 = {𝑐 ∣ ∃𝑡(𝑐 Fn 𝑡 ∧ (𝑡𝐴 ∧ ∀𝑤𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤𝑡 (𝑐𝑤) = (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤))))}
9290, 91elab2g 3670 . . 3 (𝐶 ∈ V → (𝐶𝐵 ↔ ∃𝑡(𝐶 Fn 𝑡 ∧ (𝑡𝐴 ∧ ∀𝑤𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤𝑡 (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))))
9382, 92syl 17 . 2 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → (𝐶𝐵 ↔ ∃𝑡(𝐶 Fn 𝑡 ∧ (𝑡𝐴 ∧ ∀𝑤𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤𝑡 (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))))
9475, 93mpbird 257 1 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wo 844  w3a 1086   = wceq 1540  wex 1780  wcel 2105  {cab 2708  wral 3060  Vcvv 3473  cdif 3945  cun 3946  cin 3947  wss 3948  c0 4322  {csn 4628  cop 4634   class class class wbr 5148   Fr wfr 5628  dom cdm 5676  cres 5678  Predcpred 6299  Fun wfun 6537   Fn wfn 6538  cfv 6543  (class class class)co 7412  frecscfrecs 8271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-fr 5631  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-frecs 8272
This theorem is referenced by:  frrlem14  8290
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