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Theorem frrlem13 8229
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 4086 . . . . . 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 5276 . . . . 5 (𝑆 ∈ V → (𝑆 ∩ dom 𝐹) ∈ V)
6 snex 5388 . . . . 5 {𝑧} ∈ V
7 unexg 7683 . . . . 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 8227 . . . 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 4188 . . . . . 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 3955 . . . . 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 4769 . . . . 5 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → {𝑧} ⊆ 𝐴)
2420, 23unssd 4146 . . . 4 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ⊆ 𝐴)
25 elun 4108 . . . . . . . . 9 (𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ↔ (𝑤 ∈ (𝑆 ∩ dom 𝐹) ∨ 𝑤 ∈ {𝑧}))
26 elin 3926 . . . . . . . . . 10 (𝑤 ∈ (𝑆 ∩ dom 𝐹) ↔ (𝑤𝑆𝑤 ∈ dom 𝐹))
27 velsn 4602 . . . . . . . . . 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 3230 . . . . . . . . . . . 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 8224 . . . . . . . . . . 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 4190 . . . . . . . . . 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 6283 . . . . . . . . . . . . . . . 16 Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = (Pred(𝑅, 𝐴, 𝑧) ∖ Pred(𝑅, dom 𝐹, 𝑧))
4342eqeq1i 2741 . . . . . . . . . . . . . . 15 (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ ↔ (Pred(𝑅, 𝐴, 𝑧) ∖ Pred(𝑅, dom 𝐹, 𝑧)) = ∅)
44 ssdif0 4323 . . . . . . . . . . . . . . 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 6261 . . . . . . . . . . . . . 14 Pred(𝑅, dom 𝐹, 𝑧) ⊆ dom 𝐹
4745, 46sstrdi 3956 . . . . . . . . . . . . 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 4192 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → Pred(𝑅, 𝐴, 𝑧) ⊆ (𝑆 ∩ dom 𝐹))
51 predeq3 6257 . . . . . . . . . . 11 (𝑤 = 𝑧 → Pred(𝑅, 𝐴, 𝑤) = Pred(𝑅, 𝐴, 𝑧))
5251sseq1d 3975 . . . . . . . . . 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 4132 . . . . . 6 (𝑆 ∩ dom 𝐹) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})
5856, 57sstrdi 3956 . . . . 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 3143 . . . 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 8228 . . . . . 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 3143 . . . 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 6594 . . . . . 6 (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → (𝐶 Fn 𝑡𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧})))
67 sseq1 3969 . . . . . . 7 (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → (𝑡𝐴 ↔ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ⊆ 𝐴))
68 sseq2 3970 . . . . . . . 8 (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → (Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡 ↔ Pred(𝑅, 𝐴, 𝑤) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})))
6968raleqbi1dv 3307 . . . . . . 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 3309 . . . . . 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 3556 . . . 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 8225 . . . . . 6 (𝜑 → Fun 𝐹)
77 resfunexg 7165 . . . . . 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 5388 . . . . 5 {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ∈ V
80 unexg 7683 . . . . 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 6593 . . . . . 6 (𝑐 = 𝐶 → (𝑐 Fn 𝑡𝐶 Fn 𝑡))
84 fveq1 6841 . . . . . . . 8 (𝑐 = 𝐶 → (𝑐𝑤) = (𝐶𝑤))
85 reseq1 5931 . . . . . . . . 9 (𝑐 = 𝐶 → (𝑐 ↾ Pred(𝑅, 𝐴, 𝑤)) = (𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))
8685oveq2d 7373 . . . . . . . 8 (𝑐 = 𝐶 → (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤))) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))
8784, 86eqeq12d 2752 . . . . . . 7 (𝑐 = 𝐶 → ((𝑐𝑤) = (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤))) ↔ (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))
8887ralbidv 3174 . . . . . 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 8217 . . . 4 𝐵 = {𝑐 ∣ ∃𝑡(𝑐 Fn 𝑡 ∧ (𝑡𝐴 ∧ ∀𝑤𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤𝑡 (𝑐𝑤) = (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤))))}
9290, 91elab2g 3632 . . 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 2713  wral 3064  Vcvv 3445  cdif 3907  cun 3908  cin 3909  wss 3910  c0 4282  {csn 4586  cop 4592   class class class wbr 5105   Fr wfr 5585  dom cdm 5633  cres 5635  Predcpred 6252  Fun wfun 6490   Fn wfn 6491  cfv 6496  (class class class)co 7357  frecscfrecs 8211
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-fr 5588  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-frecs 8212
This theorem is referenced by:  frrlem14  8230
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