Metamath Proof Explorer


Theorem frrlem13

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)

Ref Expression
Hypotheses frrlem11.1 𝐵 = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝑦 𝐺 ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
frrlem11.2 𝐹 = frecs ( 𝑅 , 𝐴 , 𝐺 )
frrlem11.3 ( ( 𝜑 ∧ ( 𝑔𝐵𝐵 ) ) → ( ( 𝑥 𝑔 𝑢𝑥 𝑣 ) → 𝑢 = 𝑣 ) )
frrlem11.4 𝐶 = ( ( 𝐹𝑆 ) ∪ { ⟨ 𝑧 , ( 𝑧 𝐺 ( 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 frrlem13 ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → 𝐶𝐵 )

Proof

Step Hyp Ref Expression
1 frrlem11.1 𝐵 = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝑦 𝐺 ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
2 frrlem11.2 𝐹 = frecs ( 𝑅 , 𝐴 , 𝐺 )
3 frrlem11.3 ( ( 𝜑 ∧ ( 𝑔𝐵𝐵 ) ) → ( ( 𝑥 𝑔 𝑢𝑥 𝑣 ) → 𝑢 = 𝑣 ) )
4 frrlem11.4 𝐶 = ( ( 𝐹𝑆 ) ∪ { ⟨ 𝑧 , ( 𝑧 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ⟩ } )
5 frrlem12.5 ( 𝜑𝑅 Fr 𝐴 )
6 frrlem12.6 ( ( 𝜑𝑧𝐴 ) → Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ 𝑆 )
7 frrlem12.7 ( ( 𝜑𝑧𝐴 ) → ∀ 𝑤𝑆 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑆 )
8 frrlem13.8 ( ( 𝜑𝑧𝐴 ) → 𝑆 ∈ V )
9 frrlem13.9 ( ( 𝜑𝑧𝐴 ) → 𝑆𝐴 )
10 eldifi ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) → 𝑧𝐴 )
11 10 8 sylan2 ( ( 𝜑𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ) → 𝑆 ∈ V )
12 11 adantrr ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → 𝑆 ∈ V )
13 inex1g ( 𝑆 ∈ V → ( 𝑆 ∩ dom 𝐹 ) ∈ V )
14 snex { 𝑧 } ∈ V
15 unexg ( ( ( 𝑆 ∩ dom 𝐹 ) ∈ V ∧ { 𝑧 } ∈ V ) → ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ∈ V )
16 13 14 15 sylancl ( 𝑆 ∈ V → ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ∈ V )
17 12 16 syl ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ∈ V )
18 1 2 3 4 frrlem11 ( ( 𝜑𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ) → 𝐶 Fn ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) )
19 18 adantrr ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → 𝐶 Fn ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) )
20 inss1 ( 𝑆 ∩ dom 𝐹 ) ⊆ 𝑆
21 10 9 sylan2 ( ( 𝜑𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ) → 𝑆𝐴 )
22 21 adantrr ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → 𝑆𝐴 )
23 20 22 sstrid ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → ( 𝑆 ∩ dom 𝐹 ) ⊆ 𝐴 )
24 10 adantl ( ( 𝜑𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ) → 𝑧𝐴 )
25 24 adantrr ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → 𝑧𝐴 )
26 25 snssd ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → { 𝑧 } ⊆ 𝐴 )
27 23 26 unssd ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ⊆ 𝐴 )
28 elun ( 𝑤 ∈ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ↔ ( 𝑤 ∈ ( 𝑆 ∩ dom 𝐹 ) ∨ 𝑤 ∈ { 𝑧 } ) )
29 elin ( 𝑤 ∈ ( 𝑆 ∩ dom 𝐹 ) ↔ ( 𝑤𝑆𝑤 ∈ dom 𝐹 ) )
30 velsn ( 𝑤 ∈ { 𝑧 } ↔ 𝑤 = 𝑧 )
31 29 30 orbi12i ( ( 𝑤 ∈ ( 𝑆 ∩ dom 𝐹 ) ∨ 𝑤 ∈ { 𝑧 } ) ↔ ( ( 𝑤𝑆𝑤 ∈ dom 𝐹 ) ∨ 𝑤 = 𝑧 ) )
32 28 31 bitri ( 𝑤 ∈ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ↔ ( ( 𝑤𝑆𝑤 ∈ dom 𝐹 ) ∨ 𝑤 = 𝑧 ) )
33 10 7 sylan2 ( ( 𝜑𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ) → ∀ 𝑤𝑆 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑆 )
34 33 adantrr ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → ∀ 𝑤𝑆 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑆 )
35 rsp ( ∀ 𝑤𝑆 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑆 → ( 𝑤𝑆 → Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑆 ) )
36 34 35 syl ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → ( 𝑤𝑆 → Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑆 ) )
37 1 2 frrlem8 ( 𝑤 ∈ dom 𝐹 → Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ dom 𝐹 )
38 36 37 anim12d1 ( ( 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 ) ) )
39 ssin ( ( Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑆 ∧ Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ dom 𝐹 ) ↔ Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ ( 𝑆 ∩ dom 𝐹 ) )
40 38 39 syl6ib ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → ( ( 𝑤𝑆𝑤 ∈ dom 𝐹 ) → Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ ( 𝑆 ∩ dom 𝐹 ) ) )
41 10 6 sylan2 ( ( 𝜑𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ) → Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ 𝑆 )
42 41 adantrr ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ 𝑆 )
43 preddif Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ( Pred ( 𝑅 , 𝐴 , 𝑧 ) ∖ Pred ( 𝑅 , dom 𝐹 , 𝑧 ) )
44 43 eqeq1i ( Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ↔ ( Pred ( 𝑅 , 𝐴 , 𝑧 ) ∖ Pred ( 𝑅 , dom 𝐹 , 𝑧 ) ) = ∅ )
45 ssdif0 ( Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ Pred ( 𝑅 , dom 𝐹 , 𝑧 ) ↔ ( Pred ( 𝑅 , 𝐴 , 𝑧 ) ∖ Pred ( 𝑅 , dom 𝐹 , 𝑧 ) ) = ∅ )
46 44 45 sylbb2 ( Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ → Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ Pred ( 𝑅 , dom 𝐹 , 𝑧 ) )
47 predss Pred ( 𝑅 , dom 𝐹 , 𝑧 ) ⊆ dom 𝐹
48 46 47 sstrdi ( Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ → Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ dom 𝐹 )
49 48 adantl ( ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) → Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ dom 𝐹 )
50 49 adantl ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ dom 𝐹 )
51 42 50 ssind ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ ( 𝑆 ∩ dom 𝐹 ) )
52 predeq3 ( 𝑤 = 𝑧 → Pred ( 𝑅 , 𝐴 , 𝑤 ) = Pred ( 𝑅 , 𝐴 , 𝑧 ) )
53 52 sseq1d ( 𝑤 = 𝑧 → ( Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ ( 𝑆 ∩ dom 𝐹 ) ↔ Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ ( 𝑆 ∩ dom 𝐹 ) ) )
54 51 53 syl5ibrcom ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → ( 𝑤 = 𝑧 → Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ ( 𝑆 ∩ dom 𝐹 ) ) )
55 40 54 jaod ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → ( ( ( 𝑤𝑆𝑤 ∈ dom 𝐹 ) ∨ 𝑤 = 𝑧 ) → Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ ( 𝑆 ∩ dom 𝐹 ) ) )
56 32 55 syl5bi ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → ( 𝑤 ∈ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) → Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ ( 𝑆 ∩ dom 𝐹 ) ) )
57 56 imp ( ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) ∧ 𝑤 ∈ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ) → Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ ( 𝑆 ∩ dom 𝐹 ) )
58 ssun1 ( 𝑆 ∩ dom 𝐹 ) ⊆ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } )
59 57 58 sstrdi ( ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) ∧ 𝑤 ∈ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ) → Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) )
60 59 ralrimiva ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → ∀ 𝑤 ∈ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) )
61 27 60 jca ( ( 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 } ) ) )
62 1 2 3 4 5 6 7 frrlem12 ( ( 𝜑𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ 𝑤 ∈ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ) → ( 𝐶𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) )
63 62 3expa ( ( ( 𝜑𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ) ∧ 𝑤 ∈ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ) → ( 𝐶𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) )
64 63 ralrimiva ( ( 𝜑𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ) → ∀ 𝑤 ∈ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ( 𝐶𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) )
65 64 adantrr ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → ∀ 𝑤 ∈ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ( 𝐶𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) )
66 fneq2 ( 𝑡 = ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) → ( 𝐶 Fn 𝑡𝐶 Fn ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ) )
67 sseq1 ( 𝑡 = ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) → ( 𝑡𝐴 ↔ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ⊆ 𝐴 ) )
68 sseq2 ( 𝑡 = ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) → ( Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ↔ Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ) )
69 68 raleqbi1dv ( 𝑡 = ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) → ( ∀ 𝑤𝑡 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ↔ ∀ 𝑤 ∈ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ) )
70 67 69 anbi12d ( 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 ( 𝑡 = ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) → ( ∀ 𝑤𝑡 ( 𝐶𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ↔ ∀ 𝑤 ∈ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ( 𝐶𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) )
72 66 70 71 3anbi123d ( 𝑡 = ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) → ( ( 𝐶 Fn 𝑡 ∧ ( 𝑡𝐴 ∧ ∀ 𝑤𝑡 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ) ∧ ∀ 𝑤𝑡 ( 𝐶𝑤 ) = ( 𝑤 𝐺 ( 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 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) ) )
73 72 spcegv ( ( ( 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 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) → ∃ 𝑡 ( 𝐶 Fn 𝑡 ∧ ( 𝑡𝐴 ∧ ∀ 𝑤𝑡 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ) ∧ ∀ 𝑤𝑡 ( 𝐶𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) ) )
74 73 imp ( ( ( ( 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 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) ) → ∃ 𝑡 ( 𝐶 Fn 𝑡 ∧ ( 𝑡𝐴 ∧ ∀ 𝑤𝑡 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ) ∧ ∀ 𝑤𝑡 ( 𝐶𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) )
75 17 19 61 65 74 syl13anc ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → ∃ 𝑡 ( 𝐶 Fn 𝑡 ∧ ( 𝑡𝐴 ∧ ∀ 𝑤𝑡 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ) ∧ ∀ 𝑤𝑡 ( 𝐶𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) )
76 1 2 3 frrlem9 ( 𝜑 → Fun 𝐹 )
77 resfunexg ( ( Fun 𝐹𝑆 ∈ V ) → ( 𝐹𝑆 ) ∈ V )
78 76 12 77 syl2an2r ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → ( 𝐹𝑆 ) ∈ V )
79 snex { ⟨ 𝑧 , ( 𝑧 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ⟩ } ∈ V
80 unexg ( ( ( 𝐹𝑆 ) ∈ V ∧ { ⟨ 𝑧 , ( 𝑧 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ⟩ } ∈ V ) → ( ( 𝐹𝑆 ) ∪ { ⟨ 𝑧 , ( 𝑧 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ⟩ } ) ∈ V )
81 78 79 80 sylancl ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → ( ( 𝐹𝑆 ) ∪ { ⟨ 𝑧 , ( 𝑧 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ⟩ } ) ∈ V )
82 4 81 eqeltrid ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → 𝐶 ∈ V )
83 fneq1 ( 𝑐 = 𝐶 → ( 𝑐 Fn 𝑡𝐶 Fn 𝑡 ) )
84 fveq1 ( 𝑐 = 𝐶 → ( 𝑐𝑤 ) = ( 𝐶𝑤 ) )
85 reseq1 ( 𝑐 = 𝐶 → ( 𝑐 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) = ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) )
86 85 oveq2d ( 𝑐 = 𝐶 → ( 𝑤 𝐺 ( 𝑐 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) )
87 84 86 eqeq12d ( 𝑐 = 𝐶 → ( ( 𝑐𝑤 ) = ( 𝑤 𝐺 ( 𝑐 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ↔ ( 𝐶𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) )
88 87 ralbidv ( 𝑐 = 𝐶 → ( ∀ 𝑤𝑡 ( 𝑐𝑤 ) = ( 𝑤 𝐺 ( 𝑐 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ↔ ∀ 𝑤𝑡 ( 𝐶𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) )
89 83 88 3anbi13d ( 𝑐 = 𝐶 → ( ( 𝑐 Fn 𝑡 ∧ ( 𝑡𝐴 ∧ ∀ 𝑤𝑡 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ) ∧ ∀ 𝑤𝑡 ( 𝑐𝑤 ) = ( 𝑤 𝐺 ( 𝑐 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) ↔ ( 𝐶 Fn 𝑡 ∧ ( 𝑡𝐴 ∧ ∀ 𝑤𝑡 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ) ∧ ∀ 𝑤𝑡 ( 𝐶𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) ) )
90 89 exbidv ( 𝑐 = 𝐶 → ( ∃ 𝑡 ( 𝑐 Fn 𝑡 ∧ ( 𝑡𝐴 ∧ ∀ 𝑤𝑡 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ) ∧ ∀ 𝑤𝑡 ( 𝑐𝑤 ) = ( 𝑤 𝐺 ( 𝑐 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) ↔ ∃ 𝑡 ( 𝐶 Fn 𝑡 ∧ ( 𝑡𝐴 ∧ ∀ 𝑤𝑡 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ) ∧ ∀ 𝑤𝑡 ( 𝐶𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) ) )
91 1 frrlem1 𝐵 = { 𝑐 ∣ ∃ 𝑡 ( 𝑐 Fn 𝑡 ∧ ( 𝑡𝐴 ∧ ∀ 𝑤𝑡 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ) ∧ ∀ 𝑤𝑡 ( 𝑐𝑤 ) = ( 𝑤 𝐺 ( 𝑐 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) }
92 90 91 elab2g ( 𝐶 ∈ V → ( 𝐶𝐵 ↔ ∃ 𝑡 ( 𝐶 Fn 𝑡 ∧ ( 𝑡𝐴 ∧ ∀ 𝑤𝑡 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ) ∧ ∀ 𝑤𝑡 ( 𝐶𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) ) )
93 82 92 syl ( ( 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 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) ) )
94 75 93 mpbird ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → 𝐶𝐵 )