Theorem ofcfval4 30167

Description: The function/constant operation expressed as an operation composition. (Contributed by Thierry Arnoux, 31-Jan-2017.)

Hypotheses

Ref	Expression
ofcfval4.1	|- ( ph -> F : A --> B )
ofcfval4.2	|- ( ph -> A e. V )
ofcfval4.3	|- ( ph -> C e. W )

Assertion

Ref	Expression
ofcfval4	|- ( ph -> ( F∘𝑓/𝑐 R C ) = ( ( x e. B |-> ( x R C ) ) o. F ) )

Distinct variable groups:   x, B   x, C   x, F   x, R

Allowed substitution hints:   ph( x)   A( x)   V( x)   W( x)

Proof of Theorem ofcfval4

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofcfval4.1	. . . 4 |- ( ph -> F : A --> B )
2		fdm 6051	. . . 4 |- ( F : A --> B -> dom F = A )
3	1, 2	syl 17	. . 3 |- ( ph -> dom F = A )
4	3	mpteq1d 4738	. 2 |- ( ph -> ( y e. dom F |-> ( ( F ` y ) R C ) ) = ( y e. A |-> ( ( F ` y ) R C ) ) )
5		ofcfval4.2	. . . 4 |- ( ph -> A e. V )
6		fex 6490	. . . 4 |- ( ( F : A --> B /\ A e. V ) -> F e. _V )
7	1, 5, 6	syl2anc 693	. . 3 |- ( ph -> F e. _V )
8		ofcfval4.3	. . 3 |- ( ph -> C e. W )
9		ofcfval3 30164	. . 3 |- ( ( F e. _V /\ C e. W ) -> ( F∘𝑓/𝑐 R C ) = ( y e. dom F |-> ( ( F ` y ) R C ) ) )
10	7, 8, 9	syl2anc 693	. 2 |- ( ph -> ( F∘𝑓/𝑐 R C ) = ( y e. dom F |-> ( ( F ` y ) R C ) ) )
11	1	ffvelrnda 6359	. . 3 |- ( ( ph /\ y e. A ) -> ( F ` y ) e. B )
12	1	feqmptd 6249	. . 3 |- ( ph -> F = ( y e. A |-> ( F ` y ) ) )
13		eqidd 2623	. . 3 |- ( ph -> ( x e. B |-> ( x R C ) ) = ( x e. B |-> ( x R C ) ) )
14		oveq1 6657	. . 3 |- ( x = ( F ` y ) -> ( x R C ) = ( ( F ` y ) R C ) )
15	11, 12, 13, 14	fmptco 6396	. 2 |- ( ph -> ( ( x e. B |-> ( x R C ) ) o. F ) = ( y e. A |-> ( ( F ` y ) R C ) ) )
16	4, 10, 15	3eqtr4d 2666	1 |- ( ph -> ( F∘𝑓/𝑐 R C ) = ( ( x e. B |-> ( x R C ) ) o. F ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   dom cdm 5114   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650  ∘_𝑓/𝑐cofc 30157

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  ax-un 6949

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-ofc 30158

This theorem is referenced by:  rrvmulc  30515