Theorem offveq 6296

Description: Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Mario Carneiro, 24-Jul-2014.)

Hypotheses

Ref	Expression
offveq.1	|- ( ph -> A e. V )
offveq.2	|- ( ph -> F Fn A )
offveq.3	|- ( ph -> G Fn A )
offveq.4	|- ( ph -> H Fn A )
offveq.5	|- ( ( ph /\ x e. A ) -> ( F ` x ) = B )
offveq.6	|- ( ( ph /\ x e. A ) -> ( G ` x ) = C )
offveq.7	|- ( ( ph /\ x e. A ) -> ( B R C ) = ( H ` x ) )

Assertion

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

Distinct variable groups:   x, A   x, F   x, G   x, H   ph, x   x, R

Allowed substitution hints:   B( x)   C( x)   V( x)

Proof of Theorem offveq

Step	Hyp	Ref	Expression
1		offveq.7	. . . . 5 |- ( ( ph /\ x e. A ) -> ( B R C ) = ( H ` x ) )
2	1	eqcomd 2240	. . . 4 |- ( ( ph /\ x e. A ) -> ( H ` x ) = ( B R C ) )
3	2	ralrimiva 2617	. . 3 |- ( ph -> A. x e. A ( H ` x ) = ( B R C ) )
4		offveq.1	. . . 4 |- ( ph -> A e. V )
5		offveq.2	. . . 4 |- ( ph -> F Fn A )
6		offveq.3	. . . 4 |- ( ph -> G Fn A )
7		offveq.4	. . . 4 |- ( ph -> H Fn A )
8		offveq.5	. . . 4 |- ( ( ph /\ x e. A ) -> ( F ` x ) = B )
9		offveq.6	. . . 4 |- ( ( ph /\ x e. A ) -> ( G ` x ) = C )
10	4, 5, 6, 7, 8, 9	offveqb 6295	. . 3 |- ( ph -> ( H = ( F oF R G ) <-> A. x e. A ( H ` x ) = ( B R C ) ) )
11	3, 10	mpbird 167	. 2 |- ( ph -> H = ( F oF R G ) )
12	11	eqcomd 2240	1 |- ( ph -> ( F oF R G ) = H )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   A.wral 2522   Fn wfn 5352   ` cfv 5357  (class class class)co 6058   oFcof 6273

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-of 6275

This theorem is referenced by:  caofid0l  6302  caofid0r  6303  caofid1  6304  caofid2  6305