Theorem qliftlem 6555

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 6501	. . 3 |- ( R Er X -> ( X e. _V -> R e. _V ) )
4	1, 2, 3	sylc 62	. 2 |- ( ph -> R e. _V )
5		ecelqsg 6530	. 2 |- ( ( R e. _V /\ x e. X ) -> [ x ] R e. ( X /. R ) )
6	4, 5	sylan 281	1 |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   _Vcvv 2712   <.cop 3563   |-> cmpt 4025   ran crn 4586   Er wer 6474   [cec 6475   /.cqs 6476

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4591  df-rel 4592  df-cnv 4593  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-er 6477  df-ec 6479  df-qs 6483

This theorem is referenced by:  qliftrel  6556  qliftel  6557  qliftel1  6558  qliftfun  6559  qliftf  6562  qliftval  6563