Theorem caofref 6071

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 3995	. . . 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 2539	. . . . 5 |- ( ph -> A. x e. S x R x )
5	4	adantr 274	. . . 4 |- ( ( ph /\ w e. A ) -> A. x e. S x R x )
6		caofref.2	. . . . 5 |- ( ph -> F : A --> S )
7	6	ffvelrnda 5620	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
8	2, 5, 7	rspcdva 2835	. . 3 |- ( ( ph /\ w e. A ) -> ( F ` w ) R ( F ` w ) )
9	8	ralrimiva 2539	. 2 |- ( ph -> A. w e. A ( F ` w ) R ( F ` w ) )
10		ffn 5337	. . . 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 3331	. . 3 |- ( A i^i A ) = A
14		eqidd 2166	. . 3 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
15	11, 11, 12, 12, 13, 14, 14	ofrfval 6058	. 2 |- ( ph -> ( F oR R F <-> A. w e. A ( F ` w ) R ( F ` w ) ) )
16	9, 15	mpbird 166	1 |- ( ph -> F oR R F )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   A.wral 2444   class class class wbr 3982   Fn wfn 5183   -->wf 5184   ` cfv 5188   oRcofr 6049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ofr 6051

This theorem is referenced by: (None)