Theorem caofcom 6073

Description: Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)

Hypotheses

Ref	Expression
caofref.1	|- ( ph -> A e. V )
caofref.2	|- ( ph -> F : A --> S )
caofcom.3	|- ( ph -> G : A --> S )
caofcom.4	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x R y ) = ( y R x ) )

Assertion

Ref	Expression
caofcom	|- ( ph -> ( F oF R G ) = ( G oF R F ) )

Distinct variable groups:   x, y, F   x, G, y   ph, x, y   x, R, y   x, S, y

Allowed substitution hints:   A( x, y)   V( x, y)

Proof of Theorem caofcom

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caofref.2	. . . . . 6 |- ( ph -> F : A --> S )
2	1	ffvelrnda 5620	. . . . 5 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
3		caofcom.3	. . . . . 6 |- ( ph -> G : A --> S )
4	3	ffvelrnda 5620	. . . . 5 |- ( ( ph /\ w e. A ) -> ( G ` w ) e. S )
5	2, 4	jca 304	. . . 4 |- ( ( ph /\ w e. A ) -> ( ( F ` w ) e. S /\ ( G ` w ) e. S ) )
6		caofcom.4	. . . . 5 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x R y ) = ( y R x ) )
7	6	caovcomg 5997	. . . 4 |- ( ( ph /\ ( ( F ` w ) e. S /\ ( G ` w ) e. S ) ) -> ( ( F ` w ) R ( G ` w ) ) = ( ( G ` w ) R ( F ` w ) ) )
8	5, 7	syldan 280	. . 3 |- ( ( ph /\ w e. A ) -> ( ( F ` w ) R ( G ` w ) ) = ( ( G ` w ) R ( F ` w ) ) )
9	8	mpteq2dva 4072	. 2 |- ( ph -> ( w e. A |-> ( ( F ` w ) R ( G ` w ) ) ) = ( w e. A |-> ( ( G ` w ) R ( F ` w ) ) ) )
10		caofref.1	. . 3 |- ( ph -> A e. V )
11	1	feqmptd 5539	. . 3 |- ( ph -> F = ( w e. A |-> ( F ` w ) ) )
12	3	feqmptd 5539	. . 3 |- ( ph -> G = ( w e. A |-> ( G ` w ) ) )
13	10, 2, 4, 11, 12	offval2 6065	. 2 |- ( ph -> ( F oF R G ) = ( w e. A |-> ( ( F ` w ) R ( G ` w ) ) ) )
14	10, 4, 2, 12, 11	offval2 6065	. 2 |- ( ph -> ( G oF R F ) = ( w e. A |-> ( ( G ` w ) R ( F ` w ) ) ) )
15	9, 13, 14	3eqtr4d 2208	1 |- ( ph -> ( F oF R G ) = ( G oF R F ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   |-> cmpt 4043   -->wf 5184   ` cfv 5188  (class class class)co 5842   oFcof 6048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-of 6050

This theorem is referenced by: (None)