Theorem caofid2 6928

Description: Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Hypotheses

Ref	Expression
caofref.1	|- ( ph -> A e. V )
caofref.2	|- ( ph -> F : A --> S )
caofid0.3	|- ( ph -> B e. W )
caofid1.4	|- ( ph -> C e. X )
caofid2.5	|- ( ( ph /\ x e. S ) -> ( B R x ) = C )

Assertion

Ref	Expression
caofid2	|- ( ph -> ( ( A X. { B } ) oF R F ) = ( A X. { C } ) )

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

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

Proof of Theorem caofid2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caofref.1	. 2 |- ( ph -> A e. V )
2		caofid0.3	. . 3 |- ( ph -> B e. W )
3		fnconstg 6093	. . 3 |- ( B e. W -> ( A X. { B } ) Fn A )
4	2, 3	syl 17	. 2 |- ( ph -> ( A X. { B } ) Fn A )
5		caofref.2	. . 3 |- ( ph -> F : A --> S )
6		ffn 6045	. . 3 |- ( F : A --> S -> F Fn A )
7	5, 6	syl 17	. 2 |- ( ph -> F Fn A )
8		caofid1.4	. . 3 |- ( ph -> C e. X )
9		fnconstg 6093	. . 3 |- ( C e. X -> ( A X. { C } ) Fn A )
10	8, 9	syl 17	. 2 |- ( ph -> ( A X. { C } ) Fn A )
11		fvconst2g 6467	. . 3 |- ( ( B e. W /\ w e. A ) -> ( ( A X. { B } ) ` w ) = B )
12	2, 11	sylan 488	. 2 |- ( ( ph /\ w e. A ) -> ( ( A X. { B } ) ` w ) = B )
13		eqidd 2623	. 2 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
14	5	ffvelrnda 6359	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
15		caofid2.5	. . . . . 6 |- ( ( ph /\ x e. S ) -> ( B R x ) = C )
16	15	ralrimiva 2966	. . . . 5 |- ( ph -> A. x e. S ( B R x ) = C )
17		oveq2 6658	. . . . . . 7 |- ( x = ( F ` w ) -> ( B R x ) = ( B R ( F ` w ) ) )
18	17	eqeq1d 2624	. . . . . 6 |- ( x = ( F ` w ) -> ( ( B R x ) = C <-> ( B R ( F ` w ) ) = C ) )
19	18	rspccva 3308	. . . . 5 |- ( ( A. x e. S ( B R x ) = C /\ ( F ` w ) e. S ) -> ( B R ( F ` w ) ) = C )
20	16, 19	sylan 488	. . . 4 |- ( ( ph /\ ( F ` w ) e. S ) -> ( B R ( F ` w ) ) = C )
21	14, 20	syldan 487	. . 3 |- ( ( ph /\ w e. A ) -> ( B R ( F ` w ) ) = C )
22		fvconst2g 6467	. . . 4 |- ( ( C e. X /\ w e. A ) -> ( ( A X. { C } ) ` w ) = C )
23	8, 22	sylan 488	. . 3 |- ( ( ph /\ w e. A ) -> ( ( A X. { C } ) ` w ) = C )
24	21, 23	eqtr4d 2659	. 2 |- ( ( ph /\ w e. A ) -> ( B R ( F ` w ) ) = ( ( A X. { C } ) ` w ) )
25	1, 4, 7, 10, 12, 13, 24	offveq 6918	1 |- ( ph -> ( ( A X. { B } ) oF R F ) = ( A X. { C } ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {csn 4177   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` 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:  mbfmulc2lem  23414  i1fmulc  23470  itg1mulc  23471  itg2mulc  23514  dvcmulf  23708  coe0  24012  plymul0or  24036  expgrowth  38534