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Theorem frrlem13 8221
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 4084 . . . . . 6 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝑧𝐴)
2 frrlem13.8 . . . . . 6 ((𝜑𝑧𝐴) → 𝑆 ∈ V)
31, 2sylan2 593 . . . . 5 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝑆 ∈ V)
43adantrr 715 . . . 4 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝑆 ∈ V)
5 inex1g 5274 . . . . 5 (𝑆 ∈ V → (𝑆 ∩ dom 𝐹) ∈ V)
6 snex 5386 . . . . 5 {𝑧} ∈ V
7 unexg 7675 . . . . 5 (((𝑆 ∩ dom 𝐹) ∈ V ∧ {𝑧} ∈ V) → ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ∈ V)
85, 6, 7sylancl 586 . . . 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 8219 . . . 4 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))
1514adantrr 715 . . 3 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))
16 inss1 4186 . . . . . 6 (𝑆 ∩ dom 𝐹) ⊆ 𝑆
17 frrlem13.9 . . . . . . . 8 ((𝜑𝑧𝐴) → 𝑆𝐴)
181, 17sylan2 593 . . . . . . 7 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝑆𝐴)
1918adantrr 715 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝑆𝐴)
2016, 19sstrid 3953 . . . . 5 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → (𝑆 ∩ dom 𝐹) ⊆ 𝐴)
211adantl 482 . . . . . . 7 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝑧𝐴)
2221adantrr 715 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝑧𝐴)
2322snssd 4767 . . . . 5 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → {𝑧} ⊆ 𝐴)
2420, 23unssd 4144 . . . 4 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ⊆ 𝐴)
25 elun 4106 . . . . . . . . 9 (𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ↔ (𝑤 ∈ (𝑆 ∩ dom 𝐹) ∨ 𝑤 ∈ {𝑧}))
26 elin 3924 . . . . . . . . . 10 (𝑤 ∈ (𝑆 ∩ dom 𝐹) ↔ (𝑤𝑆𝑤 ∈ dom 𝐹))
27 velsn 4600 . . . . . . . . . 10 (𝑤 ∈ {𝑧} ↔ 𝑤 = 𝑧)
2826, 27orbi12i 913 . . . . . . . . 9 ((𝑤 ∈ (𝑆 ∩ dom 𝐹) ∨ 𝑤 ∈ {𝑧}) ↔ ((𝑤𝑆𝑤 ∈ dom 𝐹) ∨ 𝑤 = 𝑧))
2925, 28bitri 274 . . . . . . . 8 (𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ↔ ((𝑤𝑆𝑤 ∈ dom 𝐹) ∨ 𝑤 = 𝑧))
30 frrlem12.7 . . . . . . . . . . . . . 14 ((𝜑𝑧𝐴) → ∀𝑤𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
311, 30sylan2 593 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ∀𝑤𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
3231adantrr 715 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ∀𝑤𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
33 rsp 3228 . . . . . . . . . . . 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 8216 . . . . . . . . . . 11 (𝑤 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹)
3634, 35anim12d1 610 . . . . . . . . . 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 4188 . . . . . . . . . 10 ((Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆 ∧ Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹) ↔ Pred(𝑅, 𝐴, 𝑤) ⊆ (𝑆 ∩ dom 𝐹))
3836, 37syl6ib 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 593 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆)
4140adantrr 715 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆)
42 preddif 6281 . . . . . . . . . . . . . . . 16 Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = (Pred(𝑅, 𝐴, 𝑧) ∖ Pred(𝑅, dom 𝐹, 𝑧))
4342eqeq1i 2742 . . . . . . . . . . . . . . 15 (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ ↔ (Pred(𝑅, 𝐴, 𝑧) ∖ Pred(𝑅, dom 𝐹, 𝑧)) = ∅)
44 ssdif0 4321 . . . . . . . . . . . . . . 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 6259 . . . . . . . . . . . . . 14 Pred(𝑅, dom 𝐹, 𝑧) ⊆ dom 𝐹
4745, 46sstrdi 3954 . . . . . . . . . . . . 13 (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
4847adantl 482 . . . . . . . . . . . 12 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
4948adantl 482 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
5041, 49ssind 4190 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → Pred(𝑅, 𝐴, 𝑧) ⊆ (𝑆 ∩ dom 𝐹))
51 predeq3 6255 . . . . . . . . . . 11 (𝑤 = 𝑧 → Pred(𝑅, 𝐴, 𝑤) = Pred(𝑅, 𝐴, 𝑧))
5251sseq1d 3973 . . . . . . . . . 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 857 . . . . . . . 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 407 . . . . . 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 4130 . . . . . 6 (𝑆 ∩ dom 𝐹) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})
5856, 57sstrdi 3954 . . . . 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 3141 . . . 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 512 . . 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 8220 . . . . . 6 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) → (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))
63623expa 1118 . . . . 5 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) → (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))
6463ralrimiva 3141 . . . 4 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})(𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))
6564adantrr 715 . . 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 6591 . . . . . 6 (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → (𝐶 Fn 𝑡𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧})))
67 sseq1 3967 . . . . . . 7 (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → (𝑡𝐴 ↔ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ⊆ 𝐴))
68 sseq2 3968 . . . . . . . 8 (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → (Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡 ↔ Pred(𝑅, 𝐴, 𝑤) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})))
6968raleqbi1dv 3305 . . . . . . 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 631 . . . . . 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 3307 . . . . . 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 1436 . . . . 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 3554 . . . 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 407 . . 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 1372 . 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 8217 . . . . . 6 (𝜑 → Fun 𝐹)
77 resfunexg 7161 . . . . . 6 ((Fun 𝐹𝑆 ∈ V) → (𝐹𝑆) ∈ V)
7876, 4, 77syl2an2r 683 . . . . 5 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → (𝐹𝑆) ∈ V)
79 snex 5386 . . . . 5 {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ∈ V
80 unexg 7675 . . . . 5 (((𝐹𝑆) ∈ V ∧ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ∈ V) → ((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ∈ V)
8178, 79, 80sylancl 586 . . . 4 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ∈ V)
8213, 81eqeltrid 2842 . . 3 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶 ∈ V)
83 fneq1 6590 . . . . . 6 (𝑐 = 𝐶 → (𝑐 Fn 𝑡𝐶 Fn 𝑡))
84 fveq1 6838 . . . . . . . 8 (𝑐 = 𝐶 → (𝑐𝑤) = (𝐶𝑤))
85 reseq1 5929 . . . . . . . . 9 (𝑐 = 𝐶 → (𝑐 ↾ Pred(𝑅, 𝐴, 𝑤)) = (𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))
8685oveq2d 7367 . . . . . . . 8 (𝑐 = 𝐶 → (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤))) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))
8784, 86eqeq12d 2753 . . . . . . 7 (𝑐 = 𝐶 → ((𝑐𝑤) = (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤))) ↔ (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))
8887ralbidv 3172 . . . . . 6 (𝑐 = 𝐶 → (∀𝑤𝑡 (𝑐𝑤) = (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤))) ↔ ∀𝑤𝑡 (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))
8983, 883anbi13d 1438 . . . . 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 1924 . . . 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 8209 . . . 4 𝐵 = {𝑐 ∣ ∃𝑡(𝑐 Fn 𝑡 ∧ (𝑡𝐴 ∧ ∀𝑤𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤𝑡 (𝑐𝑤) = (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤))))}
9290, 91elab2g 3630 . . 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 256 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 396  wo 845  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2714  wral 3062  Vcvv 3443  cdif 3905  cun 3906  cin 3907  wss 3908  c0 4280  {csn 4584  cop 4590   class class class wbr 5103   Fr wfr 5583  dom cdm 5631  cres 5633  Predcpred 6250  Fun wfun 6487   Fn wfn 6488  cfv 6493  (class class class)co 7351  frecscfrecs 8203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-fr 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-frecs 8204
This theorem is referenced by:  frrlem14  8222
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