Theorem qliftlem 6669

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 6613	. . 3 |- ( R Er X -> ( X e. _V -> R e. _V ) )
4	1, 2, 3	sylc 62	. 2 |- ( ph -> R e. _V )
5		ecelqsg 6644	. 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 2164   _Vcvv 2760   <.cop 3622   |-> cmpt 4091   ran crn 4661   Er wer 6586   [cec 6587   /.cqs 6588

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

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-v 2762  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-xp 4666  df-rel 4667  df-cnv 4668  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-er 6589  df-ec 6591  df-qs 6595

This theorem is referenced by:  qliftrel  6670  qliftel  6671  qliftel1  6672  qliftfun  6673  qliftf  6676  qliftval  6677