Theorem caofref 5757

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 5328	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
3		caofref.3	. . . . . 6 |- ( ( ph /\ x e. S ) -> x R x )
4	3	ralrimiva 2435	. . . . 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 3800	. . . . 5 |- ( x = ( F ` w ) -> ( x R x <-> ( F ` w ) R ( F ` w ) ) )
8	7	rspcv 2698	. . . 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 2435	. 2 |- ( ph -> A. w e. A ( F ` w ) R ( F ` w ) )
11		ffn 5071	. . . 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 3176	. . 3 |- ( A i^i A ) = A
15		eqidd 2083	. . 3 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
16	12, 12, 13, 13, 14, 15, 15	ofrfval 5745	. 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 1285   e. wcel 1434   A.wral 2349   class class class wbr 3787   Fn wfn 4921   -->wf 4922   ` cfv 4926   oRcofr 5736

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-pow 3950  ax-pr 3966

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-ofr 5738

This theorem is referenced by: (None)