Theorem caofcom 6156

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 5693	. . . . 5 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
3		caofcom.3	. . . . . 6 |- ( ph -> G : A --> S )
4	3	ffvelcdmda 5693	. . . . 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 6074	. . . 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 4119	. 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 5610	. . 3 |- ( ph -> F = ( w e. A |-> ( F ` w ) ) )
12	3	feqmptd 5610	. . 3 |- ( ph -> G = ( w e. A |-> ( G ` w ) ) )
13	10, 2, 4, 11, 12	offval2 6146	. 2 |- ( ph -> ( F oF R G ) = ( w e. A |-> ( ( F ` w ) R ( G ` w ) ) ) )
14	10, 4, 2, 12, 11	offval2 6146	. 2 |- ( ph -> ( G oF R F ) = ( w e. A |-> ( ( G ` w ) R ( F ` w ) ) ) )
15	9, 13, 14	3eqtr4d 2236	1 |- ( ph -> ( F oF R G ) = ( G oF R F ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   |-> cmpt 4090   -->wf 5250   ` cfv 5254  (class class class)co 5918   oFcof 6128

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-of 6130

This theorem is referenced by: (None)