Theorem offveq 6186

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 2212	. . . 4 |- ( ( ph /\ x e. A ) -> ( H ` x ) = ( B R C ) )
3	2	ralrimiva 2580	. . 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 6185	. . 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 2212	1 |- ( ph -> ( F oF R G ) = H )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   A.wral 2485   Fn wfn 5271   ` cfv 5276  (class class class)co 5951   oFcof 6163

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 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-setind 4589

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 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-ov 5954  df-oprab 5955  df-mpo 5956  df-of 6165

This theorem is referenced by:  caofid0l  6192  caofid0r  6193  caofid1  6194  caofid2  6195