Step |
Hyp |
Ref |
Expression |
1 |
|
offval2.1 |
|- ( ph -> A e. V ) |
2 |
|
offval2.2 |
|- ( ( ph /\ x e. A ) -> B e. W ) |
3 |
|
offval2.3 |
|- ( ( ph /\ x e. A ) -> C e. X ) |
4 |
|
offval2.4 |
|- ( ph -> F = ( x e. A |-> B ) ) |
5 |
|
offval2.5 |
|- ( ph -> G = ( x e. A |-> C ) ) |
6 |
2
|
ralrimiva |
|- ( ph -> A. x e. A B e. W ) |
7 |
|
eqid |
|- ( x e. A |-> B ) = ( x e. A |-> B ) |
8 |
7
|
fnmpt |
|- ( A. x e. A B e. W -> ( x e. A |-> B ) Fn A ) |
9 |
6 8
|
syl |
|- ( ph -> ( x e. A |-> B ) Fn A ) |
10 |
4
|
fneq1d |
|- ( ph -> ( F Fn A <-> ( x e. A |-> B ) Fn A ) ) |
11 |
9 10
|
mpbird |
|- ( ph -> F Fn A ) |
12 |
3
|
ralrimiva |
|- ( ph -> A. x e. A C e. X ) |
13 |
|
eqid |
|- ( x e. A |-> C ) = ( x e. A |-> C ) |
14 |
13
|
fnmpt |
|- ( A. x e. A C e. X -> ( x e. A |-> C ) Fn A ) |
15 |
12 14
|
syl |
|- ( ph -> ( x e. A |-> C ) Fn A ) |
16 |
5
|
fneq1d |
|- ( ph -> ( G Fn A <-> ( x e. A |-> C ) Fn A ) ) |
17 |
15 16
|
mpbird |
|- ( ph -> G Fn A ) |
18 |
|
inidm |
|- ( A i^i A ) = A |
19 |
4
|
adantr |
|- ( ( ph /\ y e. A ) -> F = ( x e. A |-> B ) ) |
20 |
19
|
fveq1d |
|- ( ( ph /\ y e. A ) -> ( F ` y ) = ( ( x e. A |-> B ) ` y ) ) |
21 |
5
|
adantr |
|- ( ( ph /\ y e. A ) -> G = ( x e. A |-> C ) ) |
22 |
21
|
fveq1d |
|- ( ( ph /\ y e. A ) -> ( G ` y ) = ( ( x e. A |-> C ) ` y ) ) |
23 |
11 17 1 1 18 20 22
|
ofrfval |
|- ( ph -> ( F oR R G <-> A. y e. A ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) ) ) |
24 |
|
nffvmpt1 |
|- F/_ x ( ( x e. A |-> B ) ` y ) |
25 |
|
nfcv |
|- F/_ x R |
26 |
|
nffvmpt1 |
|- F/_ x ( ( x e. A |-> C ) ` y ) |
27 |
24 25 26
|
nfbr |
|- F/ x ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) |
28 |
|
nfv |
|- F/ y ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) |
29 |
|
fveq2 |
|- ( y = x -> ( ( x e. A |-> B ) ` y ) = ( ( x e. A |-> B ) ` x ) ) |
30 |
|
fveq2 |
|- ( y = x -> ( ( x e. A |-> C ) ` y ) = ( ( x e. A |-> C ) ` x ) ) |
31 |
29 30
|
breq12d |
|- ( y = x -> ( ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) <-> ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) ) ) |
32 |
27 28 31
|
cbvralw |
|- ( A. y e. A ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) <-> A. x e. A ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) ) |
33 |
|
simpr |
|- ( ( ph /\ x e. A ) -> x e. A ) |
34 |
7
|
fvmpt2 |
|- ( ( x e. A /\ B e. W ) -> ( ( x e. A |-> B ) ` x ) = B ) |
35 |
33 2 34
|
syl2anc |
|- ( ( ph /\ x e. A ) -> ( ( x e. A |-> B ) ` x ) = B ) |
36 |
13
|
fvmpt2 |
|- ( ( x e. A /\ C e. X ) -> ( ( x e. A |-> C ) ` x ) = C ) |
37 |
33 3 36
|
syl2anc |
|- ( ( ph /\ x e. A ) -> ( ( x e. A |-> C ) ` x ) = C ) |
38 |
35 37
|
breq12d |
|- ( ( ph /\ x e. A ) -> ( ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) <-> B R C ) ) |
39 |
38
|
ralbidva |
|- ( ph -> ( A. x e. A ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) <-> A. x e. A B R C ) ) |
40 |
32 39
|
syl5bb |
|- ( ph -> ( A. y e. A ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) <-> A. x e. A B R C ) ) |
41 |
23 40
|
bitrd |
|- ( ph -> ( F oR R G <-> A. x e. A B R C ) ) |