Theorem caofref 6923

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 6359	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
3		caofref.3	. . . . . 6 |- ( ( ph /\ x e. S ) -> x R x )
4	3	ralrimiva 2966	. . . . 5 |- ( ph -> A. x e. S x R x )
5	4	adantr 481	. . . 4 |- ( ( ph /\ w e. A ) -> A. x e. S x R x )
6		id 22	. . . . . 6 |- ( x = ( F ` w ) -> x = ( F ` w ) )
7	6, 6	breq12d 4666	. . . . 5 |- ( x = ( F ` w ) -> ( x R x <-> ( F ` w ) R ( F ` w ) ) )
8	7	rspcv 3305	. . . 4 |- ( ( F ` w ) e. S -> ( A. x e. S x R x -> ( F ` w ) R ( F ` w ) ) )
9	2, 5, 8	sylc 65	. . 3 |- ( ( ph /\ w e. A ) -> ( F ` w ) R ( F ` w ) )
10	9	ralrimiva 2966	. 2 |- ( ph -> A. w e. A ( F ` w ) R ( F ` w ) )
11		ffn 6045	. . . 4 |- ( F : A --> S -> F Fn A )
12	1, 11	syl 17	. . 3 |- ( ph -> F Fn A )
13		caofref.1	. . 3 |- ( ph -> A e. V )
14		inidm 3822	. . 3 |- ( A i^i A ) = A
15		eqidd 2623	. . 3 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
16	12, 12, 13, 13, 14, 15, 15	ofrfval 6905	. 2 |- ( ph -> ( F oR R F <-> A. w e. A ( F ` w ) R ( F ` w ) ) )
17	10, 16	mpbird 247	1 |- ( ph -> F oR R F )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   Fn wfn 5883   -->wf 5884   ` cfv 5888   oRcofr 6896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ofr 6898

This theorem is referenced by:  psrridm  19404  itg2itg1  23503  itg20  23504