Theorem qliftlem 6631

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

Hypotheses

Ref	Expression
qlift.1	|- F = ran ( x e. X |-> <. [ x ] R , A >. )
qlift.2	|- ( ( ph /\ x e. X ) -> A e. Y )
qlift.3	|- ( ph -> R Er X )
qlift.4	|- ( ph -> X e. _V )

Assertion

Ref	Expression
qliftlem	|- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )

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

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

Proof of Theorem qliftlem

Step	Hyp	Ref	Expression
1		qlift.3	. . 3 |- ( ph -> R Er X )
2		qlift.4	. . 3 |- ( ph -> X e. _V )
3		erex 6577	. . 3 |- ( R Er X -> ( X e. _V -> R e. _V ) )
4	1, 2, 3	sylc 62	. 2 |- ( ph -> R e. _V )
5		ecelqsg 6606	. 2 |- ( ( R e. _V /\ x e. X ) -> [ x ] R e. ( X /. R ) )
6	4, 5	sylan 283	1 |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   _Vcvv 2752   <.cop 3610   |-> cmpt 4079   ran crn 4642   Er wer 6550   [cec 6551   /.cqs 6552

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-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4647  df-rel 4648  df-cnv 4649  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-er 6553  df-ec 6555  df-qs 6559

This theorem is referenced by:  qliftrel  6632  qliftel  6633  qliftel1  6634  qliftfun  6635  qliftf  6638  qliftval  6639