Step |
Hyp |
Ref |
Expression |
1 |
|
frrlem11.1 |
⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } |
2 |
|
frrlem11.2 |
⊢ 𝐹 = frecs ( 𝑅 , 𝐴 , 𝐺 ) |
3 |
|
frrlem11.3 |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) ) → ( ( 𝑥 𝑔 𝑢 ∧ 𝑥 ℎ 𝑣 ) → 𝑢 = 𝑣 ) ) |
4 |
|
frrlem11.4 |
⊢ 𝐶 = ( ( 𝐹 ↾ 𝑆 ) ∪ { 〈 𝑧 , ( 𝑧 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) 〉 } ) |
5 |
|
frrlem12.5 |
⊢ ( 𝜑 → 𝑅 Fr 𝐴 ) |
6 |
|
frrlem12.6 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ 𝑆 ) |
7 |
|
frrlem12.7 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ∀ 𝑤 ∈ 𝑆 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑆 ) |
8 |
|
frrlem13.8 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝑆 ∈ V ) |
9 |
|
frrlem13.9 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝑆 ⊆ 𝐴 ) |
10 |
|
eldifi |
⊢ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) → 𝑧 ∈ 𝐴 ) |
11 |
10 8
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 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 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 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 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ) → 𝑆 ⊆ 𝐴 ) |
22 |
21
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → 𝑆 ⊆ 𝐴 ) |
23 |
20 22
|
sstrid |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → ( 𝑆 ∩ dom 𝐹 ) ⊆ 𝐴 ) |
24 |
10
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ) → 𝑧 ∈ 𝐴 ) |
25 |
24
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → 𝑧 ∈ 𝐴 ) |
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 ) ↔ ( 𝑤 ∈ 𝑆 ∧ 𝑤 ∈ dom 𝐹 ) ) |
30 |
|
velsn |
⊢ ( 𝑤 ∈ { 𝑧 } ↔ 𝑤 = 𝑧 ) |
31 |
29 30
|
orbi12i |
⊢ ( ( 𝑤 ∈ ( 𝑆 ∩ dom 𝐹 ) ∨ 𝑤 ∈ { 𝑧 } ) ↔ ( ( 𝑤 ∈ 𝑆 ∧ 𝑤 ∈ dom 𝐹 ) ∨ 𝑤 = 𝑧 ) ) |
32 |
28 31
|
bitri |
⊢ ( 𝑤 ∈ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ↔ ( ( 𝑤 ∈ 𝑆 ∧ 𝑤 ∈ dom 𝐹 ) ∨ 𝑤 = 𝑧 ) ) |
33 |
10 7
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ) → ∀ 𝑤 ∈ 𝑆 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑆 ) |
34 |
33
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → ∀ 𝑤 ∈ 𝑆 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑆 ) |
35 |
|
rsp |
⊢ ( ∀ 𝑤 ∈ 𝑆 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑆 → ( 𝑤 ∈ 𝑆 → Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑆 ) ) |
36 |
34 35
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → ( 𝑤 ∈ 𝑆 → 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 ) = ∅ ) ) → ( ( 𝑤 ∈ 𝑆 ∧ 𝑤 ∈ 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 ) = ∅ ) ) → ( ( 𝑤 ∈ 𝑆 ∧ 𝑤 ∈ dom 𝐹 ) → Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ ( 𝑆 ∩ dom 𝐹 ) ) ) |
41 |
10 6
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 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 ) = ∅ ) ) → ( ( ( 𝑤 ∈ 𝑆 ∧ 𝑤 ∈ 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 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ 𝑤 ∈ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ) → ( 𝐶 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) |
63 |
62
|
3expa |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ) ∧ 𝑤 ∈ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ) → ( 𝐶 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) |
64 |
63
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ) → ∀ 𝑤 ∈ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ( 𝐶 ‘ 𝑤 ) = ( 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 } ) ( 𝐶 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) |
66 |
|
fneq2 |
⊢ ( 𝑡 = ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) → ( 𝐶 Fn 𝑡 ↔ 𝐶 Fn ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ) ) |
67 |
|
sseq1 |
⊢ ( 𝑡 = ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) → ( 𝑡 ⊆ 𝐴 ↔ ( ( 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 } ) → ( ∀ 𝑤 ∈ 𝑡 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 } ) → ( ( 𝑡 ⊆ 𝐴 ∧ ∀ 𝑤 ∈ 𝑡 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 } ) → ( ∀ 𝑤 ∈ 𝑡 ( 𝐶 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ↔ ∀ 𝑤 ∈ ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) ( 𝐶 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) ) |
72 |
66 70 71
|
3anbi123d |
⊢ ( 𝑡 = ( ( 𝑆 ∩ dom 𝐹 ) ∪ { 𝑧 } ) → ( ( 𝐶 Fn 𝑡 ∧ ( 𝑡 ⊆ 𝐴 ∧ ∀ 𝑤 ∈ 𝑡 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ) ∧ ∀ 𝑤 ∈ 𝑡 ( 𝐶 ‘ 𝑤 ) = ( 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 } ) ( 𝐶 ‘ 𝑤 ) = ( 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 } ) ( 𝐶 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) → ∃ 𝑡 ( 𝐶 Fn 𝑡 ∧ ( 𝑡 ⊆ 𝐴 ∧ ∀ 𝑤 ∈ 𝑡 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ) ∧ ∀ 𝑤 ∈ 𝑡 ( 𝐶 ‘ 𝑤 ) = ( 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 } ) ( 𝐶 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) ) → ∃ 𝑡 ( 𝐶 Fn 𝑡 ∧ ( 𝑡 ⊆ 𝐴 ∧ ∀ 𝑤 ∈ 𝑡 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ) ∧ ∀ 𝑤 ∈ 𝑡 ( 𝐶 ‘ 𝑤 ) = ( 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 ∧ ( 𝑡 ⊆ 𝐴 ∧ ∀ 𝑤 ∈ 𝑡 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ) ∧ ∀ 𝑤 ∈ 𝑡 ( 𝐶 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) ) |
76 |
1 2 3
|
frrlem9 |
⊢ ( 𝜑 → Fun 𝐹 ) |
77 |
|
resfunexg |
⊢ ( ( Fun 𝐹 ∧ 𝑆 ∈ V ) → ( 𝐹 ↾ 𝑆 ) ∈ V ) |
78 |
76 12 77
|
syl2an2r |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → ( 𝐹 ↾ 𝑆 ) ∈ V ) |
79 |
|
snex |
⊢ { 〈 𝑧 , ( 𝑧 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) 〉 } ∈ V |
80 |
|
unexg |
⊢ ( ( ( 𝐹 ↾ 𝑆 ) ∈ V ∧ { 〈 𝑧 , ( 𝑧 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) 〉 } ∈ V ) → ( ( 𝐹 ↾ 𝑆 ) ∪ { 〈 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 ) = ∅ ) ) → ( ( 𝐹 ↾ 𝑆 ) ∪ { 〈 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 𝑡 ↔ 𝐶 Fn 𝑡 ) ) |
84 |
|
fveq1 |
⊢ ( 𝑐 = 𝐶 → ( 𝑐 ‘ 𝑤 ) = ( 𝐶 ‘ 𝑤 ) ) |
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 → ( ( 𝑐 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝑐 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ↔ ( 𝐶 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) ) |
88 |
87
|
ralbidv |
⊢ ( 𝑐 = 𝐶 → ( ∀ 𝑤 ∈ 𝑡 ( 𝑐 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝑐 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ↔ ∀ 𝑤 ∈ 𝑡 ( 𝐶 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) ) |
89 |
83 88
|
3anbi13d |
⊢ ( 𝑐 = 𝐶 → ( ( 𝑐 Fn 𝑡 ∧ ( 𝑡 ⊆ 𝐴 ∧ ∀ 𝑤 ∈ 𝑡 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ) ∧ ∀ 𝑤 ∈ 𝑡 ( 𝑐 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝑐 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) ↔ ( 𝐶 Fn 𝑡 ∧ ( 𝑡 ⊆ 𝐴 ∧ ∀ 𝑤 ∈ 𝑡 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ) ∧ ∀ 𝑤 ∈ 𝑡 ( 𝐶 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) ) ) |
90 |
89
|
exbidv |
⊢ ( 𝑐 = 𝐶 → ( ∃ 𝑡 ( 𝑐 Fn 𝑡 ∧ ( 𝑡 ⊆ 𝐴 ∧ ∀ 𝑤 ∈ 𝑡 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ) ∧ ∀ 𝑤 ∈ 𝑡 ( 𝑐 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝑐 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) ↔ ∃ 𝑡 ( 𝐶 Fn 𝑡 ∧ ( 𝑡 ⊆ 𝐴 ∧ ∀ 𝑤 ∈ 𝑡 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ) ∧ ∀ 𝑤 ∈ 𝑡 ( 𝐶 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) ) ) |
91 |
1
|
frrlem1 |
⊢ 𝐵 = { 𝑐 ∣ ∃ 𝑡 ( 𝑐 Fn 𝑡 ∧ ( 𝑡 ⊆ 𝐴 ∧ ∀ 𝑤 ∈ 𝑡 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ) ∧ ∀ 𝑤 ∈ 𝑡 ( 𝑐 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝑐 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) } |
92 |
90 91
|
elab2g |
⊢ ( 𝐶 ∈ V → ( 𝐶 ∈ 𝐵 ↔ ∃ 𝑡 ( 𝐶 Fn 𝑡 ∧ ( 𝑡 ⊆ 𝐴 ∧ ∀ 𝑤 ∈ 𝑡 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ) ∧ ∀ 𝑤 ∈ 𝑡 ( 𝐶 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) ) ) |
93 |
82 92
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → ( 𝐶 ∈ 𝐵 ↔ ∃ 𝑡 ( 𝐶 Fn 𝑡 ∧ ( 𝑡 ⊆ 𝐴 ∧ ∀ 𝑤 ∈ 𝑡 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑡 ) ∧ ∀ 𝑤 ∈ 𝑡 ( 𝐶 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) ) ) |
94 |
75 93
|
mpbird |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) ) → 𝐶 ∈ 𝐵 ) |