Theorem fliftrel 5686

Description: F, a function lift, is a subset of R X. S. (Contributed by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
flift.1	|- F = ran ( x e. X |-> <. A , B >. )
flift.2	|- ( ( ph /\ x e. X ) -> A e. R )
flift.3	|- ( ( ph /\ x e. X ) -> B e. S )

Assertion

Ref	Expression
fliftrel	|- ( ph -> F C_ ( R X. S ) )

Distinct variable groups:   x, R   ph, x   x, X   x, S

Allowed substitution hints:   A( x)   B( x)   F( x)

Proof of Theorem fliftrel

Step	Hyp	Ref	Expression
1		flift.1	. 2 |- F = ran ( x e. X |-> <. A , B >. )
2		flift.2	. . . . 5 |- ( ( ph /\ x e. X ) -> A e. R )
3		flift.3	. . . . 5 |- ( ( ph /\ x e. X ) -> B e. S )
4		opelxpi 4566	. . . . 5 |- ( ( A e. R /\ B e. S ) -> <. A , B >. e. ( R X. S ) )
5	2, 3, 4	syl2anc 408	. . . 4 |- ( ( ph /\ x e. X ) -> <. A , B >. e. ( R X. S ) )
6		eqid 2137	. . . 4 |- ( x e. X |-> <. A , B >. ) = ( x e. X |-> <. A , B >. )
7	5, 6	fmptd 5567	. . 3 |- ( ph -> ( x e. X |-> <. A , B >. ) : X --> ( R X. S ) )
8		frn 5276	. . 3 |- ( ( x e. X |-> <. A , B >. ) : X --> ( R X. S ) -> ran ( x e. X |-> <. A , B >. ) C_ ( R X. S ) )
9	7, 8	syl 14	. 2 |- ( ph -> ran ( x e. X |-> <. A , B >. ) C_ ( R X. S ) )
10	1, 9	eqsstrid 3138	1 |- ( ph -> F C_ ( R X. S ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   C_ wss 3066   <.cop 3525   |-> cmpt 3984   X. cxp 4532   ran crn 4535   -->wf 5114

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126

This theorem is referenced by:  fliftcnv  5689  fliftfun  5690  fliftf  5693  qliftrel  6501