Theorem offveq 6160

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 2202	. . . 4 |- ( ( ph /\ x e. A ) -> ( H ` x ) = ( B R C ) )
3	2	ralrimiva 2570	. . 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 6159	. . 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 2202	1 |- ( ph -> ( F oF R G ) = H )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   A.wral 2475   Fn wfn 5254   ` cfv 5259  (class class class)co 5925   oFcof 6137

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-of 6139

This theorem is referenced by:  caofid0l  6166  caofid0r  6167  caofid1  6168  caofid2  6169