Theorem caofref 6331

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 4144	. . . 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 2623	. . . . 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 5846	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
8	2, 5, 7	rspcdva 2934	. . 3 |- ( ( ph /\ w e. A ) -> ( F ` w ) R ( F ` w ) )
9	8	ralrimiva 2623	. 2 |- ( ph -> A. w e. A ( F ` w ) R ( F ` w ) )
10		ffn 5536	. . . 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 3440	. . 3 |- ( A i^i A ) = A
14		eqidd 2239	. . 3 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
15	11, 11, 12, 12, 13, 14, 14	ofrfval 6315	. 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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   class class class wbr 4131   Fn wfn 5375   -->wf 5376   ` cfv 5380   oRcofr 6304

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ofr 6306

This theorem is used by:  psrridmfi  15200