Theorem caofcom 6212

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 5738	. . . . 5 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
3		caofcom.3	. . . . . 6 |- ( ph -> G : A --> S )
4	3	ffvelcdmda 5738	. . . . 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 6125	. . . 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 4150	. 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 5655	. . 3 |- ( ph -> F = ( w e. A |-> ( F ` w ) ) )
12	3	feqmptd 5655	. . 3 |- ( ph -> G = ( w e. A |-> ( G ` w ) ) )
13	10, 2, 4, 11, 12	offval2 6197	. 2 |- ( ph -> ( F oF R G ) = ( w e. A |-> ( ( F ` w ) R ( G ` w ) ) ) )
14	10, 4, 2, 12, 11	offval2 6197	. 2 |- ( ph -> ( G oF R F ) = ( w e. A |-> ( ( G ` w ) R ( F ` w ) ) ) )
15	9, 13, 14	3eqtr4d 2250	1 |- ( ph -> ( F oF R G ) = ( G oF R F ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   |-> cmpt 4121   -->wf 5286   ` cfv 5290  (class class class)co 5967   oFcof 6179

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-of 6181

This theorem is referenced by: (None)