Theorem ofcval 30161

Description: Evaluate a function/constant operation at a point. (Contributed by Thierry Arnoux, 31-Jan-2017.)

Hypotheses

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

Assertion

Ref	Expression
ofcval	|- ( ( ph /\ X e. A ) -> ( ( F∘𝑓/𝑐 R C ) ` X ) = ( B R C ) )

Proof of Theorem ofcval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofcfval.1	. . . . 5 |- ( ph -> F Fn A )
2		ofcfval.2	. . . . 5 |- ( ph -> A e. V )
3		ofcfval.3	. . . . 5 |- ( ph -> C e. W )
4		eqidd 2623	. . . . 5 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
5	1, 2, 3, 4	ofcfval 30160	. . . 4 |- ( ph -> ( F∘𝑓/𝑐 R C ) = ( x e. A |-> ( ( F ` x ) R C ) ) )
6	5	adantr 481	. . 3 |- ( ( ph /\ X e. A ) -> ( F∘𝑓/𝑐 R C ) = ( x e. A |-> ( ( F ` x ) R C ) ) )
7		simpr 477	. . . . 5 |- ( ( ( ph /\ X e. A ) /\ x = X ) -> x = X )
8	7	fveq2d 6195	. . . 4 |- ( ( ( ph /\ X e. A ) /\ x = X ) -> ( F ` x ) = ( F ` X ) )
9	8	oveq1d 6665	. . 3 |- ( ( ( ph /\ X e. A ) /\ x = X ) -> ( ( F ` x ) R C ) = ( ( F ` X ) R C ) )
10		simpr 477	. . 3 |- ( ( ph /\ X e. A ) -> X e. A )
11		ovexd 6680	. . 3 |- ( ( ph /\ X e. A ) -> ( ( F ` X ) R C ) e. _V )
12	6, 9, 10, 11	fvmptd 6288	. 2 |- ( ( ph /\ X e. A ) -> ( ( F∘𝑓/𝑐 R C ) ` X ) = ( ( F ` X ) R C ) )
13		ofcval.6	. . 3 |- ( ( ph /\ X e. A ) -> ( F ` X ) = B )
14	13	oveq1d 6665	. 2 |- ( ( ph /\ X e. A ) -> ( ( F ` X ) R C ) = ( B R C ) )
15	12, 14	eqtrd 2656	1 |- ( ( ph /\ X e. A ) -> ( ( F∘𝑓/𝑐 R C ) ` X ) = ( B R C ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   Fn wfn 5883   ` 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-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:  probfinmeasb  30491