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Theorem frrlem13 33260
 Description: Lemma for 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 4054 . . . . . 6 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝑧𝐴)
2 frrlem13.8 . . . . . 6 ((𝜑𝑧𝐴) → 𝑆 ∈ V)
31, 2sylan2 595 . . . . 5 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝑆 ∈ V)
43adantrr 716 . . . 4 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝑆 ∈ V)
5 inex1g 5187 . . . . 5 (𝑆 ∈ V → (𝑆 ∩ dom 𝐹) ∈ V)
6 snex 5297 . . . . 5 {𝑧} ∈ V
7 unexg 7454 . . . . 5 (((𝑆 ∩ dom 𝐹) ∈ V ∧ {𝑧} ∈ V) → ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ∈ V)
85, 6, 7sylancl 589 . . . 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 33258 . . . 4 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))
1514adantrr 716 . . 3 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))
16 inss1 4155 . . . . . 6 (𝑆 ∩ dom 𝐹) ⊆ 𝑆
17 frrlem13.9 . . . . . . . 8 ((𝜑𝑧𝐴) → 𝑆𝐴)
181, 17sylan2 595 . . . . . . 7 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝑆𝐴)
1918adantrr 716 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝑆𝐴)
2016, 19sstrid 3926 . . . . 5 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → (𝑆 ∩ dom 𝐹) ⊆ 𝐴)
211adantl 485 . . . . . . 7 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝑧𝐴)
2221adantrr 716 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝑧𝐴)
2322snssd 4702 . . . . 5 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → {𝑧} ⊆ 𝐴)
2420, 23unssd 4113 . . . 4 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ⊆ 𝐴)
25 elun 4076 . . . . . . . . 9 (𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ↔ (𝑤 ∈ (𝑆 ∩ dom 𝐹) ∨ 𝑤 ∈ {𝑧}))
26 elin 3897 . . . . . . . . . 10 (𝑤 ∈ (𝑆 ∩ dom 𝐹) ↔ (𝑤𝑆𝑤 ∈ dom 𝐹))
27 velsn 4541 . . . . . . . . . 10 (𝑤 ∈ {𝑧} ↔ 𝑤 = 𝑧)
2826, 27orbi12i 912 . . . . . . . . 9 ((𝑤 ∈ (𝑆 ∩ dom 𝐹) ∨ 𝑤 ∈ {𝑧}) ↔ ((𝑤𝑆𝑤 ∈ dom 𝐹) ∨ 𝑤 = 𝑧))
2925, 28bitri 278 . . . . . . . 8 (𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ↔ ((𝑤𝑆𝑤 ∈ dom 𝐹) ∨ 𝑤 = 𝑧))
30 frrlem12.7 . . . . . . . . . . . . . 14 ((𝜑𝑧𝐴) → ∀𝑤𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
311, 30sylan2 595 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ∀𝑤𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
3231adantrr 716 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ∀𝑤𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
33 rsp 3170 . . . . . . . . . . . 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 33255 . . . . . . . . . . 11 (𝑤 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹)
3634, 35anim12d1 612 . . . . . . . . . 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 4157 . . . . . . . . . 10 ((Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆 ∧ Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹) ↔ Pred(𝑅, 𝐴, 𝑤) ⊆ (𝑆 ∩ dom 𝐹))
3836, 37syl6ib 254 . . . . . . . . 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 595 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆)
4140adantrr 716 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆)
42 preddif 6141 . . . . . . . . . . . . . . . 16 Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = (Pred(𝑅, 𝐴, 𝑧) ∖ Pred(𝑅, dom 𝐹, 𝑧))
4342eqeq1i 2803 . . . . . . . . . . . . . . 15 (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ ↔ (Pred(𝑅, 𝐴, 𝑧) ∖ Pred(𝑅, dom 𝐹, 𝑧)) = ∅)
44 ssdif0 4277 . . . . . . . . . . . . . . 15 (Pred(𝑅, 𝐴, 𝑧) ⊆ Pred(𝑅, dom 𝐹, 𝑧) ↔ (Pred(𝑅, 𝐴, 𝑧) ∖ Pred(𝑅, dom 𝐹, 𝑧)) = ∅)
4543, 44sylbb2 241 . . . . . . . . . . . . . 14 (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ → Pred(𝑅, 𝐴, 𝑧) ⊆ Pred(𝑅, dom 𝐹, 𝑧))
46 predss 6123 . . . . . . . . . . . . . 14 Pred(𝑅, dom 𝐹, 𝑧) ⊆ dom 𝐹
4745, 46sstrdi 3927 . . . . . . . . . . . . 13 (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
4847adantl 485 . . . . . . . . . . . 12 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
4948adantl 485 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹)
5041, 49ssind 4159 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → Pred(𝑅, 𝐴, 𝑧) ⊆ (𝑆 ∩ dom 𝐹))
51 predeq3 6120 . . . . . . . . . . 11 (𝑤 = 𝑧 → Pred(𝑅, 𝐴, 𝑤) = Pred(𝑅, 𝐴, 𝑧))
5251sseq1d 3946 . . . . . . . . . 10 (𝑤 = 𝑧 → (Pred(𝑅, 𝐴, 𝑤) ⊆ (𝑆 ∩ dom 𝐹) ↔ Pred(𝑅, 𝐴, 𝑧) ⊆ (𝑆 ∩ dom 𝐹)))
5350, 52syl5ibrcom 250 . . . . . . . . 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, 54syl5bi 245 . . . . . . 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 410 . . . . . 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 4099 . . . . . 6 (𝑆 ∩ dom 𝐹) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})
5856, 57sstrdi 3927 . . . . 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 3149 . . . 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 515 . . 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 33259 . . . . . 6 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) → (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))
63623expa 1115 . . . . 5 (((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) → (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))
6463ralrimiva 3149 . . . 4 ((𝜑𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})(𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))
6564adantrr 716 . . 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 6415 . . . . . 6 (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → (𝐶 Fn 𝑡𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧})))
67 sseq1 3940 . . . . . . 7 (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → (𝑡𝐴 ↔ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ⊆ 𝐴))
68 sseq2 3941 . . . . . . . 8 (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → (Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡 ↔ Pred(𝑅, 𝐴, 𝑤) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})))
6968raleqbi1dv 3356 . . . . . . 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 633 . . . . . 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 3358 . . . . . 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 1433 . . . . 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 3545 . . . 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 410 . . 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 1369 . 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 33256 . . . . . 6 (𝜑 → Fun 𝐹)
77 resfunexg 6955 . . . . . 6 ((Fun 𝐹𝑆 ∈ V) → (𝐹𝑆) ∈ V)
7876, 4, 77syl2an2r 684 . . . . 5 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → (𝐹𝑆) ∈ V)
79 snex 5297 . . . . 5 {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ∈ V
80 unexg 7454 . . . . 5 (((𝐹𝑆) ∈ V ∧ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ∈ V) → ((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ∈ V)
8178, 79, 80sylancl 589 . . . 4 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ((𝐹𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ∈ V)
8213, 81eqeltrid 2894 . . 3 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶 ∈ V)
83 fneq1 6414 . . . . . 6 (𝑐 = 𝐶 → (𝑐 Fn 𝑡𝐶 Fn 𝑡))
84 fveq1 6644 . . . . . . . 8 (𝑐 = 𝐶 → (𝑐𝑤) = (𝐶𝑤))
85 reseq1 5812 . . . . . . . . 9 (𝑐 = 𝐶 → (𝑐 ↾ Pred(𝑅, 𝐴, 𝑤)) = (𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))
8685oveq2d 7151 . . . . . . . 8 (𝑐 = 𝐶 → (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤))) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))
8784, 86eqeq12d 2814 . . . . . . 7 (𝑐 = 𝐶 → ((𝑐𝑤) = (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤))) ↔ (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))
8887ralbidv 3162 . . . . . 6 (𝑐 = 𝐶 → (∀𝑤𝑡 (𝑐𝑤) = (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤))) ↔ ∀𝑤𝑡 (𝐶𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))
8983, 883anbi13d 1435 . . . . 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 1922 . . . 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 33248 . . . 4 𝐵 = {𝑐 ∣ ∃𝑡(𝑐 Fn 𝑡 ∧ (𝑡𝐴 ∧ ∀𝑤𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤𝑡 (𝑐𝑤) = (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤))))}
9290, 91elab2g 3616 . . 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 260 1 ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  ∀wral 3106  Vcvv 3441   ∖ cdif 3878   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  {csn 4525  ⟨cop 4531   class class class wbr 5030   Fr wfr 5475  dom cdm 5519   ↾ cres 5521  Predcpred 6115  Fun wfun 6318   Fn wfn 6319  ‘cfv 6324  (class class class)co 7135  frecscfrecs 33242 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-fr 5478  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-frecs 33243 This theorem is referenced by:  frrlem14  33261
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