Theorem caofref 5732

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 5302	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
3		caofref.3	. . . . . 6 |- ( ( ph /\ x e. S ) -> x R x )
4	3	ralrimiva 2392	. . . . 5 |- ( ph -> A. x e. S x R x )
5	4	adantr 261	. . . 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 3777	. . . . 5 |- ( x = ( F ` w ) -> ( x R x <-> ( F ` w ) R ( F ` w ) ) )
8	7	rspcv 2652	. . . 4 |- ( ( F ` w ) e. S -> ( A. x e. S x R x -> ( F ` w ) R ( F ` w ) ) )
9	2, 5, 8	sylc 56	. . 3 |- ( ( ph /\ w e. A ) -> ( F ` w ) R ( F ` w ) )
10	9	ralrimiva 2392	. 2 |- ( ph -> A. w e. A ( F ` w ) R ( F ` w ) )
11		ffn 5046	. . . 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 3146	. . 3 |- ( A i^i A ) = A
15		eqidd 2041	. . 3 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
16	12, 12, 13, 13, 14, 15, 15	ofrfval 5720	. 2 |- ( ph -> ( F oR R F <-> A. w e. A ( F ` w ) R ( F ` w ) ) )
17	10, 16	mpbird 156	1 |- ( ph -> F oR R F )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   A.wral 2306   class class class wbr 3764   Fn wfn 4897   -->wf 4898   ` cfv 4902   oRcofr 5711

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ofr 5713

This theorem is referenced by: (None)