Theorem qliftlem 6860

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 6804	. . 3 |- ( R Er X -> ( X e. _V -> R e. _V ) )
4	1, 2, 3	sylc 62	. 2 |- ( ph -> R e. _V )
5		ecelqsg 6835	. 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 1398   e. wcel 2205   _Vcvv 2815   <.cop 3697   |-> cmpt 4176   ran crn 4755   Er wer 6777   [cec 6778   /.cqs 6779

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-xp 4760  df-rel 4761  df-cnv 4762  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-er 6780  df-ec 6782  df-qs 6786

This theorem is referenced by:  qliftrel  6861  qliftel  6862  qliftel1  6863  qliftfun  6864  qliftf  6867  qliftval  6868