Theorem caofid2 6265

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, F   ph, x   x, R   x, S   x, C

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 5534	. . 3 |- ( B e. W -> ( A X. { B } ) Fn A )
4	2, 3	syl 14	. 2 |- ( ph -> ( A X. { B } ) Fn A )
5		caofref.2	. . 3 |- ( ph -> F : A --> S )
6	5	ffnd 5483	. 2 |- ( ph -> F Fn A )
7		caofid1.4	. . 3 |- ( ph -> C e. X )
8		fnconstg 5534	. . 3 |- ( C e. X -> ( A X. { C } ) Fn A )
9	7, 8	syl 14	. 2 |- ( ph -> ( A X. { C } ) Fn A )
10		fvconst2g 5868	. . 3 |- ( ( B e. W /\ w e. A ) -> ( ( A X. { B } ) ` w ) = B )
11	2, 10	sylan 283	. 2 |- ( ( ph /\ w e. A ) -> ( ( A X. { B } ) ` w ) = B )
12		eqidd 2232	. 2 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
13		caofid2.5	. . . . 5 |- ( ( ph /\ x e. S ) -> ( B R x ) = C )
14	13	ralrimiva 2605	. . . 4 |- ( ph -> A. x e. S ( B R x ) = C )
15	5	ffvelcdmda 5782	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
16		oveq2 6026	. . . . . 6 |- ( x = ( F ` w ) -> ( B R x ) = ( B R ( F ` w ) ) )
17	16	eqeq1d 2240	. . . . 5 |- ( x = ( F ` w ) -> ( ( B R x ) = C <-> ( B R ( F ` w ) ) = C ) )
18	17	rspccva 2909	. . . 4 |- ( ( A. x e. S ( B R x ) = C /\ ( F ` w ) e. S ) -> ( B R ( F ` w ) ) = C )
19	14, 15, 18	syl2an2r 599	. . 3 |- ( ( ph /\ w e. A ) -> ( B R ( F ` w ) ) = C )
20		fvconst2g 5868	. . . 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 2267	. 2 |- ( ( ph /\ w e. A ) -> ( B R ( F ` w ) ) = ( ( A X. { C } ) ` w ) )
23	1, 4, 6, 9, 11, 12, 22	offveq 6256	1 |- ( ph -> ( ( A X. { B } ) oF R F ) = ( A X. { C } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   A.wral 2510   {csn 3669   X. cxp 4723   Fn wfn 5321   -->wf 5322   ` cfv 5326  (class class class)co 6018   oFcof 6233

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-of 6235

This theorem is referenced by: (None)