Theorem fliftrel 6558

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 5148	. . . . 5 |- ( ( A e. R /\ B e. S ) -> <. A , B >. e. ( R X. S ) )
5	2, 3, 4	syl2anc 693	. . . 4 |- ( ( ph /\ x e. X ) -> <. A , B >. e. ( R X. S ) )
6		eqid 2622	. . . 4 |- ( x e. X |-> <. A , B >. ) = ( x e. X |-> <. A , B >. )
7	5, 6	fmptd 6385	. . 3 |- ( ph -> ( x e. X |-> <. A , B >. ) : X --> ( R X. S ) )
8		frn 6053	. . 3 |- ( ( x e. X |-> <. A , B >. ) : X --> ( R X. S ) -> ran ( x e. X |-> <. A , B >. ) C_ ( R X. S ) )
9	7, 8	syl 17	. 2 |- ( ph -> ran ( x e. X |-> <. A , B >. ) C_ ( R X. S ) )
10	1, 9	syl5eqss 3649	1 |- ( ph -> F C_ ( R X. S ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   <.cop 4183   |-> cmpt 4729   X. cxp 5112   ran crn 5115   -->wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  fliftcnv  6561  fliftfun  6562  fliftf  6565  qliftrel  7829  fmucndlem  22095  pi1xfrcnv  22857