Step |
Hyp |
Ref |
Expression |
1 |
|
brcoffn.c |
|- ( ph -> C Fn Y ) |
2 |
|
brcoffn.d |
|- ( ph -> D : X --> Y ) |
3 |
|
brcoffn.r |
|- ( ph -> A ( C o. D ) B ) |
4 |
|
fnfco |
|- ( ( C Fn Y /\ D : X --> Y ) -> ( C o. D ) Fn X ) |
5 |
1 2 4
|
syl2anc |
|- ( ph -> ( C o. D ) Fn X ) |
6 |
|
simpl |
|- ( ( ph /\ ( C o. D ) Fn X ) -> ph ) |
7 |
|
simpr |
|- ( ( ph /\ ( C o. D ) Fn X ) -> ( C o. D ) Fn X ) |
8 |
6 3
|
syl |
|- ( ( ph /\ ( C o. D ) Fn X ) -> A ( C o. D ) B ) |
9 |
|
fnbr |
|- ( ( ( C o. D ) Fn X /\ A ( C o. D ) B ) -> A e. X ) |
10 |
7 8 9
|
syl2anc |
|- ( ( ph /\ ( C o. D ) Fn X ) -> A e. X ) |
11 |
6 7 10
|
3jca |
|- ( ( ph /\ ( C o. D ) Fn X ) -> ( ph /\ ( C o. D ) Fn X /\ A e. X ) ) |
12 |
5 11
|
mpdan |
|- ( ph -> ( ph /\ ( C o. D ) Fn X /\ A e. X ) ) |
13 |
2
|
3ad2ant1 |
|- ( ( ph /\ ( C o. D ) Fn X /\ A e. X ) -> D : X --> Y ) |
14 |
|
simp3 |
|- ( ( ph /\ ( C o. D ) Fn X /\ A e. X ) -> A e. X ) |
15 |
|
fvco3 |
|- ( ( D : X --> Y /\ A e. X ) -> ( ( C o. D ) ` A ) = ( C ` ( D ` A ) ) ) |
16 |
13 14 15
|
syl2anc |
|- ( ( ph /\ ( C o. D ) Fn X /\ A e. X ) -> ( ( C o. D ) ` A ) = ( C ` ( D ` A ) ) ) |
17 |
3
|
3ad2ant1 |
|- ( ( ph /\ ( C o. D ) Fn X /\ A e. X ) -> A ( C o. D ) B ) |
18 |
|
fnbrfvb |
|- ( ( ( C o. D ) Fn X /\ A e. X ) -> ( ( ( C o. D ) ` A ) = B <-> A ( C o. D ) B ) ) |
19 |
18
|
3adant1 |
|- ( ( ph /\ ( C o. D ) Fn X /\ A e. X ) -> ( ( ( C o. D ) ` A ) = B <-> A ( C o. D ) B ) ) |
20 |
17 19
|
mpbird |
|- ( ( ph /\ ( C o. D ) Fn X /\ A e. X ) -> ( ( C o. D ) ` A ) = B ) |
21 |
16 20
|
eqtr3d |
|- ( ( ph /\ ( C o. D ) Fn X /\ A e. X ) -> ( C ` ( D ` A ) ) = B ) |
22 |
|
eqid |
|- ( D ` A ) = ( D ` A ) |
23 |
21 22
|
jctil |
|- ( ( ph /\ ( C o. D ) Fn X /\ A e. X ) -> ( ( D ` A ) = ( D ` A ) /\ ( C ` ( D ` A ) ) = B ) ) |
24 |
13
|
ffnd |
|- ( ( ph /\ ( C o. D ) Fn X /\ A e. X ) -> D Fn X ) |
25 |
|
fnbrfvb |
|- ( ( D Fn X /\ A e. X ) -> ( ( D ` A ) = ( D ` A ) <-> A D ( D ` A ) ) ) |
26 |
24 14 25
|
syl2anc |
|- ( ( ph /\ ( C o. D ) Fn X /\ A e. X ) -> ( ( D ` A ) = ( D ` A ) <-> A D ( D ` A ) ) ) |
27 |
1
|
3ad2ant1 |
|- ( ( ph /\ ( C o. D ) Fn X /\ A e. X ) -> C Fn Y ) |
28 |
13 14
|
ffvelrnd |
|- ( ( ph /\ ( C o. D ) Fn X /\ A e. X ) -> ( D ` A ) e. Y ) |
29 |
|
fnbrfvb |
|- ( ( C Fn Y /\ ( D ` A ) e. Y ) -> ( ( C ` ( D ` A ) ) = B <-> ( D ` A ) C B ) ) |
30 |
27 28 29
|
syl2anc |
|- ( ( ph /\ ( C o. D ) Fn X /\ A e. X ) -> ( ( C ` ( D ` A ) ) = B <-> ( D ` A ) C B ) ) |
31 |
26 30
|
anbi12d |
|- ( ( ph /\ ( C o. D ) Fn X /\ A e. X ) -> ( ( ( D ` A ) = ( D ` A ) /\ ( C ` ( D ` A ) ) = B ) <-> ( A D ( D ` A ) /\ ( D ` A ) C B ) ) ) |
32 |
23 31
|
mpbid |
|- ( ( ph /\ ( C o. D ) Fn X /\ A e. X ) -> ( A D ( D ` A ) /\ ( D ` A ) C B ) ) |
33 |
12 32
|
syl |
|- ( ph -> ( A D ( D ` A ) /\ ( D ` A ) C B ) ) |