Theorem fliftrel 5760

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 4636	. . . . 5 |- ( ( A e. R /\ B e. S ) -> <. A , B >. e. ( R X. S ) )
5	2, 3, 4	syl2anc 409	. . . 4 |- ( ( ph /\ x e. X ) -> <. A , B >. e. ( R X. S ) )
6		eqid 2165	. . . 4 |- ( x e. X |-> <. A , B >. ) = ( x e. X |-> <. A , B >. )
7	5, 6	fmptd 5639	. . 3 |- ( ph -> ( x e. X |-> <. A , B >. ) : X --> ( R X. S ) )
8		frn 5346	. . 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 3188	1 |- ( ph -> F C_ ( R X. S ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   C_ wss 3116   <.cop 3579   |-> cmpt 4043   X. cxp 4602   ran crn 4605   -->wf 5184

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-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-rab 2453  df-v 2728  df-sbc 2952  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-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-fv 5196

This theorem is referenced by:  fliftcnv  5763  fliftfun  5764  fliftf  5767  qliftrel  6580