Theorem offveq 6229

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 2235	. . . 4 |- ( ( ph /\ x e. A ) -> ( H ` x ) = ( B R C ) )
3	2	ralrimiva 2603	. . 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 6228	. . 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 2235	1 |- ( ph -> ( F oF R G ) = H )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   Fn wfn 5309   ` cfv 5314  (class class class)co 5994   oFcof 6206

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-setind 4626

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-ov 5997  df-oprab 5998  df-mpo 5999  df-of 6208

This theorem is referenced by:  caofid0l  6235  caofid0r  6236  caofid1  6237  caofid2  6238