Theorem caofref 6300

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		id 19	. . . . 5 |- ( x = ( F ` w ) -> x = ( F ` w ) )
2	1, 1	breq12d 4127	. . . 4 |- ( x = ( F ` w ) -> ( x R x <-> ( F ` w ) R ( F ` w ) ) )
3		caofref.3	. . . . . 6 |- ( ( ph /\ x e. S ) -> x R x )
4	3	ralrimiva 2617	. . . . 5 |- ( ph -> A. x e. S x R x )
5	4	adantr 276	. . . 4 |- ( ( ph /\ w e. A ) -> A. x e. S x R x )
6		caofref.2	. . . . 5 |- ( ph -> F : A --> S )
7	6	ffvelcdmda 5817	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
8	2, 5, 7	rspcdva 2928	. . 3 |- ( ( ph /\ w e. A ) -> ( F ` w ) R ( F ` w ) )
9	8	ralrimiva 2617	. 2 |- ( ph -> A. w e. A ( F ` w ) R ( F ` w ) )
10		ffn 5513	. . . 4 |- ( F : A --> S -> F Fn A )
11	6, 10	syl 14	. . 3 |- ( ph -> F Fn A )
12		caofref.1	. . 3 |- ( ph -> A e. V )
13		inidm 3434	. . 3 |- ( A i^i A ) = A
14		eqidd 2235	. . 3 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
15	11, 11, 12, 12, 13, 14, 14	ofrfval 6284	. 2 |- ( ph -> ( F oR R F <-> A. w e. A ( F ` w ) R ( F ` w ) ) )
16	9, 15	mpbird 167	1 |- ( ph -> F oR R F )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   A.wral 2522   class class class wbr 4114   Fn wfn 5352   -->wf 5353   ` cfv 5357   oRcofr 6274

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-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 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ofr 6276

This theorem is referenced by: (None)