Theorem offvalfv 42121

Description: The function operation expressed as a mapping with function values. (Contributed by AV, 6-Apr-2019.)

Hypotheses

Ref	Expression
offvalfv.a	|- ( ph -> A e. V )
offvalfv.f	|- ( ph -> F Fn A )
offvalfv.g	|- ( ph -> G Fn A )

Assertion

Ref	Expression
offvalfv	|- ( ph -> ( F oF R G ) = ( x e. A |-> ( ( F ` x ) R ( G ` x ) ) ) )

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

Allowed substitution hint:   V( x)

Proof of Theorem offvalfv

Step	Hyp	Ref	Expression
1		offvalfv.a	. 2 |- ( ph -> A e. V )
2		offvalfv.f	. . 3 |- ( ph -> F Fn A )
3		fnfvelrn 6356	. . 3 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. ran F )
4	2, 3	sylan 488	. 2 |- ( ( ph /\ x e. A ) -> ( F ` x ) e. ran F )
5		offvalfv.g	. . 3 |- ( ph -> G Fn A )
6		fnfvelrn 6356	. . 3 |- ( ( G Fn A /\ x e. A ) -> ( G ` x ) e. ran G )
7	5, 6	sylan 488	. 2 |- ( ( ph /\ x e. A ) -> ( G ` x ) e. ran G )
8		dffn5 6241	. . 3 |- ( F Fn A <-> F = ( x e. A |-> ( F ` x ) ) )
9	2, 8	sylib 208	. 2 |- ( ph -> F = ( x e. A |-> ( F ` x ) ) )
10		dffn5 6241	. . 3 |- ( G Fn A <-> G = ( x e. A |-> ( G ` x ) ) )
11	5, 10	sylib 208	. 2 |- ( ph -> G = ( x e. A |-> ( G ` x ) ) )
12	1, 4, 7, 9, 11	offval2 6914	1 |- ( ph -> ( F oF R G ) = ( x e. A |-> ( ( F ` x ) R ( G ` x ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   |-> cmpt 4729   ran crn 5115   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-8 1992  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-pow 4843  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:  zlmodzxzscm  42135  zlmodzxzadd  42136  mndpsuppss  42152  lincsum  42218