Theorem ofcof 30169

Description: Relate function operation with operation with a constant. (Contributed by Thierry Arnoux, 3-Oct-2018.)

Hypotheses

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

Assertion

Ref	Expression
ofcof	|- ( ph -> ( F∘𝑓/𝑐 R C ) = ( F oF R ( A X. { C } ) ) )

Proof of Theorem ofcof

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofcof.1	. . . 4 |- ( ph -> F : A --> B )
2		ffn 6045	. . . 4 |- ( F : A --> B -> F Fn A )
3	1, 2	syl 17	. . 3 |- ( ph -> F Fn A )
4		ofcof.2	. . 3 |- ( ph -> A e. V )
5		ofcof.3	. . 3 |- ( ph -> C e. W )
6		eqidd 2623	. . 3 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
7	3, 4, 5, 6	ofcfval 30160	. 2 |- ( ph -> ( F∘𝑓/𝑐 R C ) = ( x e. A |-> ( ( F ` x ) R C ) ) )
8		fnconstg 6093	. . . 4 |- ( C e. W -> ( A X. { C } ) Fn A )
9	5, 8	syl 17	. . 3 |- ( ph -> ( A X. { C } ) Fn A )
10		inidm 3822	. . 3 |- ( A i^i A ) = A
11		fvconst2g 6467	. . . 4 |- ( ( C e. W /\ x e. A ) -> ( ( A X. { C } ) ` x ) = C )
12	5, 11	sylan 488	. . 3 |- ( ( ph /\ x e. A ) -> ( ( A X. { C } ) ` x ) = C )
13	3, 9, 4, 4, 10, 6, 12	offval 6904	. 2 |- ( ph -> ( F oF R ( A X. { C } ) ) = ( x e. A |-> ( ( F ` x ) R C ) ) )
14	7, 13	eqtr4d 2659	1 |- ( ph -> ( F∘𝑓/𝑐 R C ) = ( F oF R ( A X. { C } ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   |-> cmpt 4729   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895  ∘_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-of 6897  df-ofc 30158

This theorem is referenced by:  ofcccat  30620