Theorem caofref 5752

Description: Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Hypotheses

Ref	Expression
caofref.1	|- ( ph -> A e. V )
caofref.2	|- ( ph -> F : A --> S )
caofref.3	|- ( ( ph /\ x e. S ) -> x R x )

Assertion

Ref	Expression
caofref	|- ( ph -> F oR R F )

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

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

Proof of Theorem caofref

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caofref.2	. . . . 5 |- ( ph -> F : A --> S )
2	1	ffvelrnda 5323	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
3		caofref.3	. . . . . 6 |- ( ( ph /\ x e. S ) -> x R x )
4	3	ralrimiva 2434	. . . . 5 |- ( ph -> A. x e. S x R x )
5	4	adantr 270	. . . 4 |- ( ( ph /\ w e. A ) -> A. x e. S x R x )
6		id 19	. . . . . 6 |- ( x = ( F ` w ) -> x = ( F ` w ) )
7	6, 6	breq12d 3798	. . . . 5 |- ( x = ( F ` w ) -> ( x R x <-> ( F ` w ) R ( F ` w ) ) )
8	7	rspcv 2697	. . . 4 |- ( ( F ` w ) e. S -> ( A. x e. S x R x -> ( F ` w ) R ( F ` w ) ) )
9	2, 5, 8	sylc 61	. . 3 |- ( ( ph /\ w e. A ) -> ( F ` w ) R ( F ` w ) )
10	9	ralrimiva 2434	. 2 |- ( ph -> A. w e. A ( F ` w ) R ( F ` w ) )
11		ffn 5066	. . . 4 |- ( F : A --> S -> F Fn A )
12	1, 11	syl 14	. . 3 |- ( ph -> F Fn A )
13		caofref.1	. . 3 |- ( ph -> A e. V )
14		inidm 3175	. . 3 |- ( A i^i A ) = A
15		eqidd 2082	. . 3 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
16	12, 12, 13, 13, 14, 15, 15	ofrfval 5740	. 2 |- ( ph -> ( F oR R F <-> A. w e. A ( F ` w ) R ( F ` w ) ) )
17	10, 16	mpbird 165	1 |- ( ph -> F oR R F )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   A.wral 2348   class class class wbr 3785   Fn wfn 4917   -->wf 4918   ` cfv 4922   oRcofr 5731

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ofr 5733

This theorem is referenced by: (None)