Step |
Hyp |
Ref |
Expression |
1 |
|
relres |
⊢ Rel ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) |
2 |
1
|
a1i |
⊢ ( ( ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ∧ Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) → Rel ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) ) |
3 |
|
dmfco |
⊢ ( ( Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺 ) → ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ↔ ( 𝐺 ‘ 𝑋 ) ∈ dom 𝐹 ) ) |
4 |
3
|
biimpd |
⊢ ( ( Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺 ) → ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) → ( 𝐺 ‘ 𝑋 ) ∈ dom 𝐹 ) ) |
5 |
4
|
funfni |
⊢ ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) → ( 𝐺 ‘ 𝑋 ) ∈ dom 𝐹 ) ) |
6 |
|
dmressnsn |
⊢ ( ( 𝐺 ‘ 𝑋 ) ∈ dom 𝐹 → dom ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) = { ( 𝐺 ‘ 𝑋 ) } ) |
7 |
|
eleq2 |
⊢ ( dom ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) = { ( 𝐺 ‘ 𝑋 ) } → ( 𝑥 ∈ dom ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) ↔ 𝑥 ∈ { ( 𝐺 ‘ 𝑋 ) } ) ) |
8 |
|
velsn |
⊢ ( 𝑥 ∈ { ( 𝐺 ‘ 𝑋 ) } ↔ 𝑥 = ( 𝐺 ‘ 𝑋 ) ) |
9 |
|
dmressnsn |
⊢ ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) → dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) = { 𝑋 } ) |
10 |
|
dffun7 |
⊢ ( Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ↔ ( Rel ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ∧ ∀ 𝑥 ∈ dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ∃* 𝑦 𝑥 ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) 𝑦 ) ) |
11 |
|
snidg |
⊢ ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) → 𝑋 ∈ { 𝑋 } ) |
12 |
11
|
adantl |
⊢ ( ( dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) = { 𝑋 } ∧ 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ) → 𝑋 ∈ { 𝑋 } ) |
13 |
|
eleq2 |
⊢ ( { 𝑋 } = dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) → ( 𝑋 ∈ { 𝑋 } ↔ 𝑋 ∈ dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ) |
14 |
13
|
eqcoms |
⊢ ( dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) = { 𝑋 } → ( 𝑋 ∈ { 𝑋 } ↔ 𝑋 ∈ dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ) |
15 |
14
|
adantr |
⊢ ( ( dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) = { 𝑋 } ∧ 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ) → ( 𝑋 ∈ { 𝑋 } ↔ 𝑋 ∈ dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ) |
16 |
12 15
|
mpbid |
⊢ ( ( dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) = { 𝑋 } ∧ 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ) → 𝑋 ∈ dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) |
17 |
|
fvex |
⊢ ( 𝐺 ‘ 𝑋 ) ∈ V |
18 |
17
|
isseti |
⊢ ∃ 𝑧 𝑧 = ( 𝐺 ‘ 𝑋 ) |
19 |
|
eqcom |
⊢ ( 𝑧 = ( 𝐺 ‘ 𝑋 ) ↔ ( 𝐺 ‘ 𝑋 ) = 𝑧 ) |
20 |
|
fnbrfvb |
⊢ ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ( ( 𝐺 ‘ 𝑋 ) = 𝑧 ↔ 𝑋 𝐺 𝑧 ) ) |
21 |
19 20
|
syl5bb |
⊢ ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ( 𝑧 = ( 𝐺 ‘ 𝑋 ) ↔ 𝑋 𝐺 𝑧 ) ) |
22 |
21
|
biimpd |
⊢ ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ( 𝑧 = ( 𝐺 ‘ 𝑋 ) → 𝑋 𝐺 𝑧 ) ) |
23 |
|
breq1 |
⊢ ( ( 𝐺 ‘ 𝑋 ) = 𝑧 → ( ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ↔ 𝑧 𝐹 𝑦 ) ) |
24 |
23
|
eqcoms |
⊢ ( 𝑧 = ( 𝐺 ‘ 𝑋 ) → ( ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ↔ 𝑧 𝐹 𝑦 ) ) |
25 |
24
|
biimpcd |
⊢ ( ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 → ( 𝑧 = ( 𝐺 ‘ 𝑋 ) → 𝑧 𝐹 𝑦 ) ) |
26 |
22 25
|
anim12ii |
⊢ ( ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) → ( 𝑧 = ( 𝐺 ‘ 𝑋 ) → ( 𝑋 𝐺 𝑧 ∧ 𝑧 𝐹 𝑦 ) ) ) |
27 |
26
|
eximdv |
⊢ ( ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) → ( ∃ 𝑧 𝑧 = ( 𝐺 ‘ 𝑋 ) → ∃ 𝑧 ( 𝑋 𝐺 𝑧 ∧ 𝑧 𝐹 𝑦 ) ) ) |
28 |
18 27
|
mpi |
⊢ ( ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) → ∃ 𝑧 ( 𝑋 𝐺 𝑧 ∧ 𝑧 𝐹 𝑦 ) ) |
29 |
|
simpr |
⊢ ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → 𝑋 ∈ 𝐴 ) |
30 |
|
vex |
⊢ 𝑦 ∈ V |
31 |
|
brcog |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ V ) → ( 𝑋 ( 𝐹 ∘ 𝐺 ) 𝑦 ↔ ∃ 𝑧 ( 𝑋 𝐺 𝑧 ∧ 𝑧 𝐹 𝑦 ) ) ) |
32 |
29 30 31
|
sylancl |
⊢ ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ( 𝑋 ( 𝐹 ∘ 𝐺 ) 𝑦 ↔ ∃ 𝑧 ( 𝑋 𝐺 𝑧 ∧ 𝑧 𝐹 𝑦 ) ) ) |
33 |
32
|
adantr |
⊢ ( ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) → ( 𝑋 ( 𝐹 ∘ 𝐺 ) 𝑦 ↔ ∃ 𝑧 ( 𝑋 𝐺 𝑧 ∧ 𝑧 𝐹 𝑦 ) ) ) |
34 |
28 33
|
mpbird |
⊢ ( ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) → 𝑋 ( 𝐹 ∘ 𝐺 ) 𝑦 ) |
35 |
30
|
brresi |
⊢ ( 𝑋 ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) 𝑦 ↔ ( 𝑋 ∈ { 𝑋 } ∧ 𝑋 ( 𝐹 ∘ 𝐺 ) 𝑦 ) ) |
36 |
|
snidg |
⊢ ( 𝑋 ∈ 𝐴 → 𝑋 ∈ { 𝑋 } ) |
37 |
36
|
biantrurd |
⊢ ( 𝑋 ∈ 𝐴 → ( 𝑋 ( 𝐹 ∘ 𝐺 ) 𝑦 ↔ ( 𝑋 ∈ { 𝑋 } ∧ 𝑋 ( 𝐹 ∘ 𝐺 ) 𝑦 ) ) ) |
38 |
35 37
|
bitr4id |
⊢ ( 𝑋 ∈ 𝐴 → ( 𝑋 ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) 𝑦 ↔ 𝑋 ( 𝐹 ∘ 𝐺 ) 𝑦 ) ) |
39 |
38
|
ad2antlr |
⊢ ( ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) → ( 𝑋 ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) 𝑦 ↔ 𝑋 ( 𝐹 ∘ 𝐺 ) 𝑦 ) ) |
40 |
34 39
|
mpbird |
⊢ ( ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) → 𝑋 ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) 𝑦 ) |
41 |
40
|
ex |
⊢ ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ( ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 → 𝑋 ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) 𝑦 ) ) |
42 |
41
|
adantl |
⊢ ( ( ( ( dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) = { 𝑋 } ∧ 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ) ∧ 𝑥 = 𝑋 ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) → ( ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 → 𝑋 ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) 𝑦 ) ) |
43 |
|
breq1 |
⊢ ( 𝑋 = 𝑥 → ( 𝑋 ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) 𝑦 ↔ 𝑥 ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) 𝑦 ) ) |
44 |
43
|
eqcoms |
⊢ ( 𝑥 = 𝑋 → ( 𝑋 ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) 𝑦 ↔ 𝑥 ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) 𝑦 ) ) |
45 |
44
|
ad2antlr |
⊢ ( ( ( ( dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) = { 𝑋 } ∧ 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ) ∧ 𝑥 = 𝑋 ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) → ( 𝑋 ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) 𝑦 ↔ 𝑥 ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) 𝑦 ) ) |
46 |
42 45
|
sylibd |
⊢ ( ( ( ( dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) = { 𝑋 } ∧ 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ) ∧ 𝑥 = 𝑋 ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) → ( ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 → 𝑥 ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) 𝑦 ) ) |
47 |
46
|
moimdv |
⊢ ( ( ( ( dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) = { 𝑋 } ∧ 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ) ∧ 𝑥 = 𝑋 ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) → ( ∃* 𝑦 𝑥 ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) 𝑦 → ∃* 𝑦 ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) ) |
48 |
47
|
ex |
⊢ ( ( ( dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) = { 𝑋 } ∧ 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ) ∧ 𝑥 = 𝑋 ) → ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ( ∃* 𝑦 𝑥 ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) 𝑦 → ∃* 𝑦 ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) ) ) |
49 |
48
|
com23 |
⊢ ( ( ( dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) = { 𝑋 } ∧ 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ) ∧ 𝑥 = 𝑋 ) → ( ∃* 𝑦 𝑥 ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) 𝑦 → ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ∃* 𝑦 ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) ) ) |
50 |
16 49
|
rspcimdv |
⊢ ( ( dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) = { 𝑋 } ∧ 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ) → ( ∀ 𝑥 ∈ dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ∃* 𝑦 𝑥 ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) 𝑦 → ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ∃* 𝑦 ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) ) ) |
51 |
50
|
ex |
⊢ ( dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) = { 𝑋 } → ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) → ( ∀ 𝑥 ∈ dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ∃* 𝑦 𝑥 ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) 𝑦 → ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ∃* 𝑦 ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) ) ) ) |
52 |
51
|
com13 |
⊢ ( ∀ 𝑥 ∈ dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ∃* 𝑦 𝑥 ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) 𝑦 → ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) → ( dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) = { 𝑋 } → ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ∃* 𝑦 ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) ) ) ) |
53 |
10 52
|
simplbiim |
⊢ ( Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) → ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) → ( dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) = { 𝑋 } → ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ∃* 𝑦 ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) ) ) ) |
54 |
53
|
com13 |
⊢ ( dom ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) = { 𝑋 } → ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) → ( Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) → ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ∃* 𝑦 ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) ) ) ) |
55 |
9 54
|
mpcom |
⊢ ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) → ( Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) → ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ∃* 𝑦 ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) ) ) |
56 |
55
|
imp31 |
⊢ ( ( ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ∧ Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) → ∃* 𝑦 ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) |
57 |
17
|
snid |
⊢ ( 𝐺 ‘ 𝑋 ) ∈ { ( 𝐺 ‘ 𝑋 ) } |
58 |
57
|
biantrur |
⊢ ( ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ↔ ( ( 𝐺 ‘ 𝑋 ) ∈ { ( 𝐺 ‘ 𝑋 ) } ∧ ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) ) |
59 |
58
|
a1i |
⊢ ( ( ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ∧ Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) → ( ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ↔ ( ( 𝐺 ‘ 𝑋 ) ∈ { ( 𝐺 ‘ 𝑋 ) } ∧ ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) ) ) |
60 |
59
|
mobidv |
⊢ ( ( ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ∧ Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) → ( ∃* 𝑦 ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ↔ ∃* 𝑦 ( ( 𝐺 ‘ 𝑋 ) ∈ { ( 𝐺 ‘ 𝑋 ) } ∧ ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) ) ) |
61 |
56 60
|
mpbid |
⊢ ( ( ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ∧ Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) → ∃* 𝑦 ( ( 𝐺 ‘ 𝑋 ) ∈ { ( 𝐺 ‘ 𝑋 ) } ∧ ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) ) |
62 |
61
|
adantl |
⊢ ( ( 𝑥 = ( 𝐺 ‘ 𝑋 ) ∧ ( ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ∧ Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) ) → ∃* 𝑦 ( ( 𝐺 ‘ 𝑋 ) ∈ { ( 𝐺 ‘ 𝑋 ) } ∧ ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) ) |
63 |
|
breq1 |
⊢ ( 𝑥 = ( 𝐺 ‘ 𝑋 ) → ( 𝑥 ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) 𝑦 ↔ ( 𝐺 ‘ 𝑋 ) ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) 𝑦 ) ) |
64 |
30
|
brresi |
⊢ ( ( 𝐺 ‘ 𝑋 ) ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) 𝑦 ↔ ( ( 𝐺 ‘ 𝑋 ) ∈ { ( 𝐺 ‘ 𝑋 ) } ∧ ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) ) |
65 |
63 64
|
bitr2di |
⊢ ( 𝑥 = ( 𝐺 ‘ 𝑋 ) → ( ( ( 𝐺 ‘ 𝑋 ) ∈ { ( 𝐺 ‘ 𝑋 ) } ∧ ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) ↔ 𝑥 ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) 𝑦 ) ) |
66 |
65
|
adantr |
⊢ ( ( 𝑥 = ( 𝐺 ‘ 𝑋 ) ∧ ( ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ∧ Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) ) → ( ( ( 𝐺 ‘ 𝑋 ) ∈ { ( 𝐺 ‘ 𝑋 ) } ∧ ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) ↔ 𝑥 ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) 𝑦 ) ) |
67 |
66
|
mobidv |
⊢ ( ( 𝑥 = ( 𝐺 ‘ 𝑋 ) ∧ ( ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ∧ Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) ) → ( ∃* 𝑦 ( ( 𝐺 ‘ 𝑋 ) ∈ { ( 𝐺 ‘ 𝑋 ) } ∧ ( 𝐺 ‘ 𝑋 ) 𝐹 𝑦 ) ↔ ∃* 𝑦 𝑥 ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) 𝑦 ) ) |
68 |
62 67
|
mpbid |
⊢ ( ( 𝑥 = ( 𝐺 ‘ 𝑋 ) ∧ ( ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ∧ Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) ) → ∃* 𝑦 𝑥 ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) 𝑦 ) |
69 |
68
|
ex |
⊢ ( 𝑥 = ( 𝐺 ‘ 𝑋 ) → ( ( ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ∧ Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) → ∃* 𝑦 𝑥 ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) 𝑦 ) ) |
70 |
8 69
|
sylbi |
⊢ ( 𝑥 ∈ { ( 𝐺 ‘ 𝑋 ) } → ( ( ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ∧ Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) → ∃* 𝑦 𝑥 ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) 𝑦 ) ) |
71 |
7 70
|
syl6bi |
⊢ ( dom ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) = { ( 𝐺 ‘ 𝑋 ) } → ( 𝑥 ∈ dom ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) → ( ( ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ∧ Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) → ∃* 𝑦 𝑥 ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) 𝑦 ) ) ) |
72 |
71
|
com23 |
⊢ ( dom ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) = { ( 𝐺 ‘ 𝑋 ) } → ( ( ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ∧ Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) → ( 𝑥 ∈ dom ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) → ∃* 𝑦 𝑥 ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) 𝑦 ) ) ) |
73 |
6 72
|
syl |
⊢ ( ( 𝐺 ‘ 𝑋 ) ∈ dom 𝐹 → ( ( ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ∧ Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) → ( 𝑥 ∈ dom ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) → ∃* 𝑦 𝑥 ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) 𝑦 ) ) ) |
74 |
5 73
|
syl6com |
⊢ ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) → ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ( ( ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ∧ Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) → ( 𝑥 ∈ dom ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) → ∃* 𝑦 𝑥 ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) 𝑦 ) ) ) ) |
75 |
74
|
a1d |
⊢ ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) → ( Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) → ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ( ( ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ∧ Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) → ( 𝑥 ∈ dom ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) → ∃* 𝑦 𝑥 ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) 𝑦 ) ) ) ) ) |
76 |
75
|
imp31 |
⊢ ( ( ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ∧ Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) → ( ( ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ∧ Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) → ( 𝑥 ∈ dom ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) → ∃* 𝑦 𝑥 ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) 𝑦 ) ) ) |
77 |
76
|
pm2.43i |
⊢ ( ( ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ∧ Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) → ( 𝑥 ∈ dom ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) → ∃* 𝑦 𝑥 ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) 𝑦 ) ) |
78 |
77
|
ralrimiv |
⊢ ( ( ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ∧ Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) → ∀ 𝑥 ∈ dom ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) ∃* 𝑦 𝑥 ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) 𝑦 ) |
79 |
|
dffun7 |
⊢ ( Fun ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) ↔ ( Rel ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) ∧ ∀ 𝑥 ∈ dom ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) ∃* 𝑦 𝑥 ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) 𝑦 ) ) |
80 |
2 78 79
|
sylanbrc |
⊢ ( ( ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ∧ Fun ( ( 𝐹 ∘ 𝐺 ) ↾ { 𝑋 } ) ) ∧ ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) → Fun ( 𝐹 ↾ { ( 𝐺 ‘ 𝑋 ) } ) ) |