Theorem fnfvof 6911

Description: Function value of a pointwise composition. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Jun-2015.)

Assertion

Ref	Expression
fnfvof	|- ( ( ( F Fn A /\ G Fn A ) /\ ( A e. V /\ X e. A ) ) -> ( ( F oF R G ) ` X ) = ( ( F ` X ) R ( G ` X ) ) )

Proof of Theorem fnfvof

Step	Hyp	Ref	Expression
1		simpll 790	. . 3 |- ( ( ( F Fn A /\ G Fn A ) /\ A e. V ) -> F Fn A )
2		simplr 792	. . 3 |- ( ( ( F Fn A /\ G Fn A ) /\ A e. V ) -> G Fn A )
3		simpr 477	. . 3 |- ( ( ( F Fn A /\ G Fn A ) /\ A e. V ) -> A e. V )
4		inidm 3822	. . 3 |- ( A i^i A ) = A
5		eqidd 2623	. . 3 |- ( ( ( ( F Fn A /\ G Fn A ) /\ A e. V ) /\ X e. A ) -> ( F ` X ) = ( F ` X ) )
6		eqidd 2623	. . 3 |- ( ( ( ( F Fn A /\ G Fn A ) /\ A e. V ) /\ X e. A ) -> ( G ` X ) = ( G ` X ) )
7	1, 2, 3, 3, 4, 5, 6	ofval 6906	. 2 |- ( ( ( ( F Fn A /\ G Fn A ) /\ A e. V ) /\ X e. A ) -> ( ( F oF R G ) ` X ) = ( ( F ` X ) R ( G ` X ) ) )
8	7	anasss 679	1 |- ( ( ( F Fn A /\ G Fn A ) /\ ( A e. V /\ X e. A ) ) -> ( ( F oF R G ) ` X ) = ( ( F ` X ) R ( G ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   oFcof 6895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897

This theorem is referenced by:  ofccat  13708  ghmplusg  18249  lcomfsupp  18903  lmhmplusg  19044  evlslem3  19514  evlslem1  19515  coe1addfv  19635  evl1addd  19705  evl1subd  19706  evl1muld  19707  frlmvscaval  20110  frlmsslsp  20135  frlmup1  20137  frlmup2  20138  islindf4  20177  mamudi  20209  mamudir  20210  mdetrlin  20408  nmotri  22543  mdegaddle  23834  ply1rem  23923  fta1glem2  23926  fta1blem  23928  plyexmo  24068  ulmdvlem1  24154  jensen  24715  dchrmulcl  24974  dchrinv  24986  sumdchr2  24995  dchr2sum  24998  mzpsubst  37311  mzpcong  37539  rngunsnply  37743  lincsum  42218