Theorem fliftrel 5998

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 4806	. . . . 5 |- ( ( A e. R /\ B e. S ) -> <. A , B >. e. ( R X. S ) )
5	2, 3, 4	syl2anc 415	. . . 4 |- ( ( ph /\ x e. X ) -> <. A , B >. e. ( R X. S ) )
6		eqid 2238	. . . 4 |- ( x e. X |-> <. A , B >. ) = ( x e. X |-> <. A , B >. )
7	5, 6	fmptd 5862	. . 3 |- ( ph -> ( x e. X |-> <. A , B >. ) : X --> ( R X. S ) )
8		frn 5542	. . 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 3294	1 |- ( ph -> F C_ ( R X. S ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   C_ wss 3220   <.cop 3712   |-> cmpt 4192   X. cxp 4772   ran crn 4775   -->wf 5373

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-sep 4249  ax-pow 4311  ax-pr 4346

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-rab 2537  df-v 2823  df-sbc 3052  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-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385

This theorem is used by:  fliftcnv  6001  fliftfun  6002  fliftf  6005  qliftrel  6888