Theorem caofcom 5734

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 5302	. . . . 5 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
3		caofcom.3	. . . . . 6 |- ( ph -> G : A --> S )
4	3	ffvelrnda 5302	. . . . 5 |- ( ( ph /\ w e. A ) -> ( G ` w ) e. S )
5	2, 4	jca 290	. . . 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 5656	. . . 4 |- ( ( ph /\ ( ( F ` w ) e. S /\ ( G ` w ) e. S ) ) -> ( ( F ` w ) R ( G ` w ) ) = ( ( G ` w ) R ( F ` w ) ) )
8	5, 7	syldan 266	. . 3 |- ( ( ph /\ w e. A ) -> ( ( F ` w ) R ( G ` w ) ) = ( ( G ` w ) R ( F ` w ) ) )
9	8	mpteq2dva 3847	. 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 5226	. . 3 |- ( ph -> F = ( w e. A |-> ( F ` w ) ) )
12	3	feqmptd 5226	. . 3 |- ( ph -> G = ( w e. A |-> ( G ` w ) ) )
13	10, 2, 4, 11, 12	offval2 5726	. 2 |- ( ph -> ( F oF R G ) = ( w e. A |-> ( ( F ` w ) R ( G ` w ) ) ) )
14	10, 4, 2, 12, 11	offval2 5726	. 2 |- ( ph -> ( G oF R F ) = ( w e. A |-> ( ( G ` w ) R ( F ` w ) ) ) )
15	9, 13, 14	3eqtr4d 2082	1 |- ( ph -> ( F oF R G ) = ( G oF R F ) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   |-> cmpt 3818   -->wf 4898   ` cfv 4902  (class class class)co 5512   oFcof 5710

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-of 5712

This theorem is referenced by: (None)