Theorem caofid1 6331

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 5534	. 2 |- ( ph -> F Fn A )
4		caofid0.3	. . 3 |- ( ph -> B e. W )
5		fnconstg 5590	. . 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 5590	. . 3 |- ( C e. X -> ( A X. { C } ) Fn A )
9	7, 8	syl 14	. 2 |- ( ph -> ( A X. { C } ) Fn A )
10		eqidd 2239	. 2 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
11		fvconst2g 5929	. . 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 2623	. . . 4 |- ( ph -> A. x e. S ( x R B ) = C )
15	2	ffvelcdmda 5843	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
16		oveq1 6092	. . . . . 6 |- ( x = ( F ` w ) -> ( x R B ) = ( ( F ` w ) R B ) )
17	16	eqeq1d 2247	. . . . 5 |- ( x = ( F ` w ) -> ( ( x R B ) = C <-> ( ( F ` w ) R B ) = C ) )
18	17	rspccva 2928	. . . 4 |- ( ( A. x e. S ( x R B ) = C /\ ( F ` w ) e. S ) -> ( ( F ` w ) R B ) = C )
19	14, 15, 18	syl2an2r 603	. . 3 |- ( ( ph /\ w e. A ) -> ( ( F ` w ) R B ) = C )
20		fvconst2g 5929	. . . 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 2274	. 2 |- ( ( ph /\ w e. A ) -> ( ( F ` w ) R B ) = ( ( A X. { C } ) ` w ) )
23	1, 3, 6, 9, 10, 12, 22	offveq 6323	1 |- ( ph -> ( F oF R ( A X. { B } ) ) = ( A X. { C } ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   {csn 3709   X. cxp 4772   Fn wfn 5372   -->wf 5373   ` cfv 5377  (class class class)co 6085   oFcof 6300

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302

This theorem is used by: (None)