Step |
Hyp |
Ref |
Expression |
1 |
|
wfrlem13OLD.1 |
⊢ 𝑅 We 𝐴 |
2 |
|
wfrlem13OLD.2 |
⊢ 𝑅 Se 𝐴 |
3 |
|
wfrlem13OLD.3 |
⊢ 𝐹 = wrecs ( 𝑅 , 𝐴 , 𝐺 ) |
4 |
|
wfrlem13OLD.4 |
⊢ 𝐶 = ( 𝐹 ∪ { 〈 𝑧 , ( 𝐺 ‘ ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) 〉 } ) |
5 |
1 2 3 4
|
wfrlem13OLD |
⊢ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) → 𝐶 Fn ( dom 𝐹 ∪ { 𝑧 } ) ) |
6 |
5
|
adantr |
⊢ ( ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) → 𝐶 Fn ( dom 𝐹 ∪ { 𝑧 } ) ) |
7 |
1 3
|
wfrlem10OLD |
⊢ ( ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) → Pred ( 𝑅 , 𝐴 , 𝑧 ) = dom 𝐹 ) |
8 |
|
eldifi |
⊢ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) → 𝑧 ∈ 𝐴 ) |
9 |
|
setlikespec |
⊢ ( ( 𝑧 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) → Pred ( 𝑅 , 𝐴 , 𝑧 ) ∈ V ) |
10 |
8 2 9
|
sylancl |
⊢ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) → Pred ( 𝑅 , 𝐴 , 𝑧 ) ∈ V ) |
11 |
10
|
adantr |
⊢ ( ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) → Pred ( 𝑅 , 𝐴 , 𝑧 ) ∈ V ) |
12 |
7 11
|
eqeltrrd |
⊢ ( ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) → dom 𝐹 ∈ V ) |
13 |
|
snex |
⊢ { 𝑧 } ∈ V |
14 |
|
unexg |
⊢ ( ( dom 𝐹 ∈ V ∧ { 𝑧 } ∈ V ) → ( dom 𝐹 ∪ { 𝑧 } ) ∈ V ) |
15 |
13 14
|
mpan2 |
⊢ ( dom 𝐹 ∈ V → ( dom 𝐹 ∪ { 𝑧 } ) ∈ V ) |
16 |
|
fnex |
⊢ ( ( 𝐶 Fn ( dom 𝐹 ∪ { 𝑧 } ) ∧ ( dom 𝐹 ∪ { 𝑧 } ) ∈ V ) → 𝐶 ∈ V ) |
17 |
15 16
|
sylan2 |
⊢ ( ( 𝐶 Fn ( dom 𝐹 ∪ { 𝑧 } ) ∧ dom 𝐹 ∈ V ) → 𝐶 ∈ V ) |
18 |
6 12 17
|
syl2anc |
⊢ ( ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) → 𝐶 ∈ V ) |
19 |
12 13 14
|
sylancl |
⊢ ( ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) → ( dom 𝐹 ∪ { 𝑧 } ) ∈ V ) |
20 |
3
|
wfrdmssOLD |
⊢ dom 𝐹 ⊆ 𝐴 |
21 |
8
|
snssd |
⊢ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) → { 𝑧 } ⊆ 𝐴 ) |
22 |
|
unss |
⊢ ( ( dom 𝐹 ⊆ 𝐴 ∧ { 𝑧 } ⊆ 𝐴 ) ↔ ( dom 𝐹 ∪ { 𝑧 } ) ⊆ 𝐴 ) |
23 |
22
|
biimpi |
⊢ ( ( dom 𝐹 ⊆ 𝐴 ∧ { 𝑧 } ⊆ 𝐴 ) → ( dom 𝐹 ∪ { 𝑧 } ) ⊆ 𝐴 ) |
24 |
20 21 23
|
sylancr |
⊢ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) → ( dom 𝐹 ∪ { 𝑧 } ) ⊆ 𝐴 ) |
25 |
24
|
adantr |
⊢ ( ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) → ( dom 𝐹 ∪ { 𝑧 } ) ⊆ 𝐴 ) |
26 |
|
elun |
⊢ ( 𝑦 ∈ ( dom 𝐹 ∪ { 𝑧 } ) ↔ ( 𝑦 ∈ dom 𝐹 ∨ 𝑦 ∈ { 𝑧 } ) ) |
27 |
|
velsn |
⊢ ( 𝑦 ∈ { 𝑧 } ↔ 𝑦 = 𝑧 ) |
28 |
27
|
orbi2i |
⊢ ( ( 𝑦 ∈ dom 𝐹 ∨ 𝑦 ∈ { 𝑧 } ) ↔ ( 𝑦 ∈ dom 𝐹 ∨ 𝑦 = 𝑧 ) ) |
29 |
26 28
|
bitri |
⊢ ( 𝑦 ∈ ( dom 𝐹 ∪ { 𝑧 } ) ↔ ( 𝑦 ∈ dom 𝐹 ∨ 𝑦 = 𝑧 ) ) |
30 |
3
|
wfrdmclOLD |
⊢ ( 𝑦 ∈ dom 𝐹 → Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ dom 𝐹 ) |
31 |
|
ssun3 |
⊢ ( Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ dom 𝐹 → Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ ( dom 𝐹 ∪ { 𝑧 } ) ) |
32 |
30 31
|
syl |
⊢ ( 𝑦 ∈ dom 𝐹 → Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ ( dom 𝐹 ∪ { 𝑧 } ) ) |
33 |
32
|
a1i |
⊢ ( ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) → ( 𝑦 ∈ dom 𝐹 → Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ ( dom 𝐹 ∪ { 𝑧 } ) ) ) |
34 |
|
ssun1 |
⊢ dom 𝐹 ⊆ ( dom 𝐹 ∪ { 𝑧 } ) |
35 |
7 34
|
eqsstrdi |
⊢ ( ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) → Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ ( dom 𝐹 ∪ { 𝑧 } ) ) |
36 |
|
predeq3 |
⊢ ( 𝑦 = 𝑧 → Pred ( 𝑅 , 𝐴 , 𝑦 ) = Pred ( 𝑅 , 𝐴 , 𝑧 ) ) |
37 |
36
|
sseq1d |
⊢ ( 𝑦 = 𝑧 → ( Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ ( dom 𝐹 ∪ { 𝑧 } ) ↔ Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ ( dom 𝐹 ∪ { 𝑧 } ) ) ) |
38 |
35 37
|
syl5ibrcom |
⊢ ( ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) → ( 𝑦 = 𝑧 → Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ ( dom 𝐹 ∪ { 𝑧 } ) ) ) |
39 |
33 38
|
jaod |
⊢ ( ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) → ( ( 𝑦 ∈ dom 𝐹 ∨ 𝑦 = 𝑧 ) → Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ ( dom 𝐹 ∪ { 𝑧 } ) ) ) |
40 |
29 39
|
syl5bi |
⊢ ( ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) → ( 𝑦 ∈ ( dom 𝐹 ∪ { 𝑧 } ) → Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ ( dom 𝐹 ∪ { 𝑧 } ) ) ) |
41 |
40
|
ralrimiv |
⊢ ( ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) → ∀ 𝑦 ∈ ( dom 𝐹 ∪ { 𝑧 } ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ ( dom 𝐹 ∪ { 𝑧 } ) ) |
42 |
25 41
|
jca |
⊢ ( ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) → ( ( dom 𝐹 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ∀ 𝑦 ∈ ( dom 𝐹 ∪ { 𝑧 } ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ ( dom 𝐹 ∪ { 𝑧 } ) ) ) |
43 |
1 2 3 4
|
wfrlem14OLD |
⊢ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) → ( 𝑦 ∈ ( dom 𝐹 ∪ { 𝑧 } ) → ( 𝐶 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) |
44 |
43
|
ralrimiv |
⊢ ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) → ∀ 𝑦 ∈ ( dom 𝐹 ∪ { 𝑧 } ) ( 𝐶 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) |
45 |
44
|
adantr |
⊢ ( ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) → ∀ 𝑦 ∈ ( dom 𝐹 ∪ { 𝑧 } ) ( 𝐶 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) |
46 |
6 42 45
|
3jca |
⊢ ( ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) → ( 𝐶 Fn ( dom 𝐹 ∪ { 𝑧 } ) ∧ ( ( dom 𝐹 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ∀ 𝑦 ∈ ( dom 𝐹 ∪ { 𝑧 } ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ ( dom 𝐹 ∪ { 𝑧 } ) ) ∧ ∀ 𝑦 ∈ ( dom 𝐹 ∪ { 𝑧 } ) ( 𝐶 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) |
47 |
|
fneq2 |
⊢ ( 𝑥 = ( dom 𝐹 ∪ { 𝑧 } ) → ( 𝐶 Fn 𝑥 ↔ 𝐶 Fn ( dom 𝐹 ∪ { 𝑧 } ) ) ) |
48 |
|
sseq1 |
⊢ ( 𝑥 = ( dom 𝐹 ∪ { 𝑧 } ) → ( 𝑥 ⊆ 𝐴 ↔ ( dom 𝐹 ∪ { 𝑧 } ) ⊆ 𝐴 ) ) |
49 |
|
sseq2 |
⊢ ( 𝑥 = ( dom 𝐹 ∪ { 𝑧 } ) → ( Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ↔ Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ ( dom 𝐹 ∪ { 𝑧 } ) ) ) |
50 |
49
|
raleqbi1dv |
⊢ ( 𝑥 = ( dom 𝐹 ∪ { 𝑧 } ) → ( ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ↔ ∀ 𝑦 ∈ ( dom 𝐹 ∪ { 𝑧 } ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ ( dom 𝐹 ∪ { 𝑧 } ) ) ) |
51 |
48 50
|
anbi12d |
⊢ ( 𝑥 = ( dom 𝐹 ∪ { 𝑧 } ) → ( ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ↔ ( ( dom 𝐹 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ∀ 𝑦 ∈ ( dom 𝐹 ∪ { 𝑧 } ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ ( dom 𝐹 ∪ { 𝑧 } ) ) ) ) |
52 |
|
raleq |
⊢ ( 𝑥 = ( dom 𝐹 ∪ { 𝑧 } ) → ( ∀ 𝑦 ∈ 𝑥 ( 𝐶 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ↔ ∀ 𝑦 ∈ ( dom 𝐹 ∪ { 𝑧 } ) ( 𝐶 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) |
53 |
47 51 52
|
3anbi123d |
⊢ ( 𝑥 = ( dom 𝐹 ∪ { 𝑧 } ) → ( ( 𝐶 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐶 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ↔ ( 𝐶 Fn ( dom 𝐹 ∪ { 𝑧 } ) ∧ ( ( dom 𝐹 ∪ { 𝑧 } ) ⊆ 𝐴 ∧ ∀ 𝑦 ∈ ( dom 𝐹 ∪ { 𝑧 } ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ ( dom 𝐹 ∪ { 𝑧 } ) ) ∧ ∀ 𝑦 ∈ ( dom 𝐹 ∪ { 𝑧 } ) ( 𝐶 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) ) |
54 |
19 46 53
|
spcedv |
⊢ ( ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) → ∃ 𝑥 ( 𝐶 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐶 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) |
55 |
|
fneq1 |
⊢ ( 𝑓 = 𝐶 → ( 𝑓 Fn 𝑥 ↔ 𝐶 Fn 𝑥 ) ) |
56 |
|
fveq1 |
⊢ ( 𝑓 = 𝐶 → ( 𝑓 ‘ 𝑦 ) = ( 𝐶 ‘ 𝑦 ) ) |
57 |
|
reseq1 |
⊢ ( 𝑓 = 𝐶 → ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) = ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) |
58 |
57
|
fveq2d |
⊢ ( 𝑓 = 𝐶 → ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) = ( 𝐺 ‘ ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) |
59 |
56 58
|
eqeq12d |
⊢ ( 𝑓 = 𝐶 → ( ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ↔ ( 𝐶 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) |
60 |
59
|
ralbidv |
⊢ ( 𝑓 = 𝐶 → ( ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ↔ ∀ 𝑦 ∈ 𝑥 ( 𝐶 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) |
61 |
55 60
|
3anbi13d |
⊢ ( 𝑓 = 𝐶 → ( ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ↔ ( 𝐶 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐶 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) ) |
62 |
61
|
exbidv |
⊢ ( 𝑓 = 𝐶 → ( ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ↔ ∃ 𝑥 ( 𝐶 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐶 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝐶 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) ) |
63 |
18 54 62
|
elabd |
⊢ ( ( 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) ∧ Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ ) → 𝐶 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) |