Theorem caofid2 6274

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 5543	. . 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 5490	. 2 |- ( ph -> F Fn A )
7		caofid1.4	. . 3 |- ( ph -> C e. X )
8		fnconstg 5543	. . 3 |- ( C e. X -> ( A X. { C } ) Fn A )
9	7, 8	syl 14	. 2 |- ( ph -> ( A X. { C } ) Fn A )
10		fvconst2g 5876	. . 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 2606	. . . 4 |- ( ph -> A. x e. S ( B R x ) = C )
15	5	ffvelcdmda 5790	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
16		oveq2 6036	. . . . . 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 2910	. . . 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 5876	. . . 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 6265	1 |- ( ph -> ( ( A X. { B } ) oF R F ) = ( A X. { C } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   A.wral 2511   {csn 3673   X. cxp 4729   Fn wfn 5328   -->wf 5329   ` cfv 5333  (class class class)co 6028   oFcof 6242

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-of 6244

This theorem is referenced by: (None)