Theorem caofid1 6168

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 5411	. 2 |- ( ph -> F Fn A )
4		caofid0.3	. . 3 |- ( ph -> B e. W )
5		fnconstg 5458	. . 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 5458	. . 3 |- ( C e. X -> ( A X. { C } ) Fn A )
9	7, 8	syl 14	. 2 |- ( ph -> ( A X. { C } ) Fn A )
10		eqidd 2197	. 2 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
11		fvconst2g 5779	. . 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 2570	. . . 4 |- ( ph -> A. x e. S ( x R B ) = C )
15	2	ffvelcdmda 5700	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
16		oveq1 5932	. . . . . 6 |- ( x = ( F ` w ) -> ( x R B ) = ( ( F ` w ) R B ) )
17	16	eqeq1d 2205	. . . . 5 |- ( x = ( F ` w ) -> ( ( x R B ) = C <-> ( ( F ` w ) R B ) = C ) )
18	17	rspccva 2867	. . . 4 |- ( ( A. x e. S ( x R B ) = C /\ ( F ` w ) e. S ) -> ( ( F ` w ) R B ) = C )
19	14, 15, 18	syl2an2r 595	. . 3 |- ( ( ph /\ w e. A ) -> ( ( F ` w ) R B ) = C )
20		fvconst2g 5779	. . . 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 2232	. 2 |- ( ( ph /\ w e. A ) -> ( ( F ` w ) R B ) = ( ( A X. { C } ) ` w ) )
23	1, 3, 6, 9, 10, 12, 22	offveq 6160	1 |- ( ph -> ( F oF R ( A X. { B } ) ) = ( A X. { C } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   A.wral 2475   {csn 3623   X. cxp 4662   Fn wfn 5254   -->wf 5255   ` cfv 5259  (class class class)co 5925   oFcof 6137

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-of 6139

This theorem is referenced by: (None)