Theorem ofcfval 30160

Description: Value of an operation applied to a function and a constant. (Contributed by Thierry Arnoux, 30-Jan-2017.)

Hypotheses

Ref	Expression
ofcfval.1	|- ( ph -> F Fn A )
ofcfval.2	|- ( ph -> A e. V )
ofcfval.3	|- ( ph -> C e. W )
ofcfval.6	|- ( ( ph /\ x e. A ) -> ( F ` x ) = B )

Assertion

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

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

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

Proof of Theorem ofcfval

Dummy variables f c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ofc 30158	. . . 4 |- ∘𝑓/𝑐 R = ( f e. _V , c e. _V |-> ( x e. dom f |-> ( ( f ` x ) R c ) ) )
2	1	a1i 11	. . 3 |- ( ph -> ∘𝑓/𝑐 R = ( f e. _V , c e. _V |-> ( x e. dom f |-> ( ( f ` x ) R c ) ) ) )
3		simprl 794	. . . . 5 |- ( ( ph /\ ( f = F /\ c = C ) ) -> f = F )
4	3	dmeqd 5326	. . . 4 |- ( ( ph /\ ( f = F /\ c = C ) ) -> dom f = dom F )
5	3	fveq1d 6193	. . . . 5 |- ( ( ph /\ ( f = F /\ c = C ) ) -> ( f ` x ) = ( F ` x ) )
6		simprr 796	. . . . 5 |- ( ( ph /\ ( f = F /\ c = C ) ) -> c = C )
7	5, 6	oveq12d 6668	. . . 4 |- ( ( ph /\ ( f = F /\ c = C ) ) -> ( ( f ` x ) R c ) = ( ( F ` x ) R C ) )
8	4, 7	mpteq12dv 4733	. . 3 |- ( ( ph /\ ( f = F /\ c = C ) ) -> ( x e. dom f |-> ( ( f ` x ) R c ) ) = ( x e. dom F |-> ( ( F ` x ) R C ) ) )
9		ofcfval.1	. . . 4 |- ( ph -> F Fn A )
10		ofcfval.2	. . . 4 |- ( ph -> A e. V )
11		fnex 6481	. . . 4 |- ( ( F Fn A /\ A e. V ) -> F e. _V )
12	9, 10, 11	syl2anc 693	. . 3 |- ( ph -> F e. _V )
13		ofcfval.3	. . . 4 |- ( ph -> C e. W )
14		elex 3212	. . . 4 |- ( C e. W -> C e. _V )
15	13, 14	syl 17	. . 3 |- ( ph -> C e. _V )
16		fndm 5990	. . . . . 6 |- ( F Fn A -> dom F = A )
17	9, 16	syl 17	. . . . 5 |- ( ph -> dom F = A )
18	17, 10	eqeltrd 2701	. . . 4 |- ( ph -> dom F e. V )
19		mptexg 6484	. . . 4 |- ( dom F e. V -> ( x e. dom F |-> ( ( F ` x ) R C ) ) e. _V )
20	18, 19	syl 17	. . 3 |- ( ph -> ( x e. dom F |-> ( ( F ` x ) R C ) ) e. _V )
21	2, 8, 12, 15, 20	ovmpt2d 6788	. 2 |- ( ph -> ( F∘𝑓/𝑐 R C ) = ( x e. dom F |-> ( ( F ` x ) R C ) ) )
22	17	eleq2d 2687	. . . . . 6 |- ( ph -> ( x e. dom F <-> x e. A ) )
23	22	pm5.32i 669	. . . . 5 |- ( ( ph /\ x e. dom F ) <-> ( ph /\ x e. A ) )
24		ofcfval.6	. . . . 5 |- ( ( ph /\ x e. A ) -> ( F ` x ) = B )
25	23, 24	sylbi 207	. . . 4 |- ( ( ph /\ x e. dom F ) -> ( F ` x ) = B )
26	25	oveq1d 6665	. . 3 |- ( ( ph /\ x e. dom F ) -> ( ( F ` x ) R C ) = ( B R C ) )
27	17, 26	mpteq12dva 4732	. 2 |- ( ph -> ( x e. dom F |-> ( ( F ` x ) R C ) ) = ( x e. A |-> ( B R C ) ) )
28	21, 27	eqtrd 2656	1 |- ( ph -> ( F∘𝑓/𝑐 R C ) = ( x e. A |-> ( B R C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   dom cdm 5114   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652  ∘_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-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-ofc 30158

This theorem is referenced by:  ofcval  30161  ofcfn  30162  ofcfeqd2  30163  ofcf  30165  ofcfval2  30166  ofcc  30168  ofcof  30169