Theorem fliftrel 5836

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 4692	. . . . 5 |- ( ( A e. R /\ B e. S ) -> <. A , B >. e. ( R X. S ) )
5	2, 3, 4	syl2anc 411	. . . 4 |- ( ( ph /\ x e. X ) -> <. A , B >. e. ( R X. S ) )
6		eqid 2193	. . . 4 |- ( x e. X |-> <. A , B >. ) = ( x e. X |-> <. A , B >. )
7	5, 6	fmptd 5713	. . 3 |- ( ph -> ( x e. X |-> <. A , B >. ) : X --> ( R X. S ) )
8		frn 5413	. . 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 3226	1 |- ( ph -> F C_ ( R X. S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   C_ wss 3154   <.cop 3622   |-> cmpt 4091   X. cxp 4658   ran crn 4661   -->wf 5251

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263

This theorem is referenced by:  fliftcnv  5839  fliftfun  5840  fliftf  5843  qliftrel  6670