Theorem caofref 6159

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 4046	. . . 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 2570	. . . . 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 5697	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
8	2, 5, 7	rspcdva 2873	. . 3 |- ( ( ph /\ w e. A ) -> ( F ` w ) R ( F ` w ) )
9	8	ralrimiva 2570	. 2 |- ( ph -> A. w e. A ( F ` w ) R ( F ` w ) )
10		ffn 5407	. . . 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 3372	. . 3 |- ( A i^i A ) = A
14		eqidd 2197	. . 3 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
15	11, 11, 12, 12, 13, 14, 14	ofrfval 6144	. 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 1364   e. wcel 2167   A.wral 2475   class class class wbr 4033   Fn wfn 5253   -->wf 5254   ` cfv 5258   oRcofr 6134

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ofr 6136

This theorem is referenced by: (None)