Theorem caofcom 6323

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	ffvelcdmda 5834	. . . . 5 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
3		caofcom.3	. . . . . 6 |- ( ph -> G : A --> S )
4	3	ffvelcdmda 5834	. . . . 5 |- ( ( ph /\ w e. A ) -> ( G ` w ) e. S )
5	2, 4	jca 306	. . . 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 6235	. . . 4 |- ( ( ph /\ ( ( F ` w ) e. S /\ ( G ` w ) e. S ) ) -> ( ( F ` w ) R ( G ` w ) ) = ( ( G ` w ) R ( F ` w ) ) )
8	5, 7	syldan 282	. . 3 |- ( ( ph /\ w e. A ) -> ( ( F ` w ) R ( G ` w ) ) = ( ( G ` w ) R ( F ` w ) ) )
9	8	mpteq2dva 4216	. 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 5750	. . 3 |- ( ph -> F = ( w e. A |-> ( F ` w ) ) )
12	3	feqmptd 5750	. . 3 |- ( ph -> G = ( w e. A |-> ( G ` w ) ) )
13	10, 2, 4, 11, 12	offval2 6308	. 2 |- ( ph -> ( F oF R G ) = ( w e. A |-> ( ( F ` w ) R ( G ` w ) ) ) )
14	10, 4, 2, 12, 11	offval2 6308	. 2 |- ( ph -> ( G oF R F ) = ( w e. A |-> ( ( G ` w ) R ( F ` w ) ) ) )
15	9, 13, 14	3eqtr4d 2281	1 |- ( ph -> ( F oF R G ) = ( G oF R F ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   |-> cmpt 4187   -->wf 5368   ` cfv 5372  (class class class)co 6075   oFcof 6290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-of 6292

This theorem is referenced by: (None)