Theorem caofid1 6253

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 )
caofid1.5	|- ( ( ph /\ x e. S ) -> ( x R B ) = C )

Assertion

Ref	Expression
caofid1	|- ( ph -> ( F oF R ( A X. { B } ) ) = ( A X. { C } ) )

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

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

Proof of Theorem caofid1

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caofref.1	. 2 |- ( ph -> A e. V )
2		caofref.2	. . 3 |- ( ph -> F : A --> S )
3	2	ffnd 5474	. 2 |- ( ph -> F Fn A )
4		caofid0.3	. . 3 |- ( ph -> B e. W )
5		fnconstg 5525	. . 3 |- ( B e. W -> ( A X. { B } ) Fn A )
6	4, 5	syl 14	. 2 |- ( ph -> ( A X. { B } ) Fn A )
7		caofid1.4	. . 3 |- ( ph -> C e. X )
8		fnconstg 5525	. . 3 |- ( C e. X -> ( A X. { C } ) Fn A )
9	7, 8	syl 14	. 2 |- ( ph -> ( A X. { C } ) Fn A )
10		eqidd 2230	. 2 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
11		fvconst2g 5857	. . 3 |- ( ( B e. W /\ w e. A ) -> ( ( A X. { B } ) ` w ) = B )
12	4, 11	sylan 283	. 2 |- ( ( ph /\ w e. A ) -> ( ( A X. { B } ) ` w ) = B )
13		caofid1.5	. . . . 5 |- ( ( ph /\ x e. S ) -> ( x R B ) = C )
14	13	ralrimiva 2603	. . . 4 |- ( ph -> A. x e. S ( x R B ) = C )
15	2	ffvelcdmda 5772	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
16		oveq1 6014	. . . . . 6 |- ( x = ( F ` w ) -> ( x R B ) = ( ( F ` w ) R B ) )
17	16	eqeq1d 2238	. . . . 5 |- ( x = ( F ` w ) -> ( ( x R B ) = C <-> ( ( F ` w ) R B ) = C ) )
18	17	rspccva 2906	. . . 4 |- ( ( A. x e. S ( x R B ) = C /\ ( F ` w ) e. S ) -> ( ( F ` w ) R B ) = C )
19	14, 15, 18	syl2an2r 597	. . 3 |- ( ( ph /\ w e. A ) -> ( ( F ` w ) R B ) = C )
20		fvconst2g 5857	. . . 4 |- ( ( C e. X /\ w e. A ) -> ( ( A X. { C } ) ` w ) = C )
21	7, 20	sylan 283	. . 3 |- ( ( ph /\ w e. A ) -> ( ( A X. { C } ) ` w ) = C )
22	19, 21	eqtr4d 2265	. 2 |- ( ( ph /\ w e. A ) -> ( ( F ` w ) R B ) = ( ( A X. { C } ) ` w ) )
23	1, 3, 6, 9, 10, 12, 22	offveq 6245	1 |- ( ph -> ( F oF R ( A X. { B } ) ) = ( A X. { C } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   {csn 3666   X. cxp 4717   Fn wfn 5313   -->wf 5314   ` cfv 5318  (class class class)co 6007   oFcof 6222

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-of 6224

This theorem is referenced by: (None)