Theorem qliftel1 6210

Description: Elementhood in the relation F. (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
qliftel1	|- ( ( ph /\ x e. X ) -> [ x ] R F A )

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

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

Proof of Theorem qliftel1

Step	Hyp	Ref	Expression
1		qlift.1	. 2 |- F = ran ( x e. X |-> <. [ x ] R , A >. )
2		qlift.2	. . 3 |- ( ( ph /\ x e. X ) -> A e. Y )
3		qlift.3	. . 3 |- ( ph -> R Er X )
4		qlift.4	. . 3 |- ( ph -> X e. _V )
5	1, 2, 3, 4	qliftlem 6207	. 2 |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )
6	1, 5, 2	fliftel1 5454	1 |- ( ( ph /\ x e. X ) -> [ x ] R F A )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   _Vcvv 2601   <.cop 3401   class class class wbr 3785   |-> cmpt 3839   ran crn 4364   Er wer 6126   [cec 6127   /.cqs 6128

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-xp 4369  df-rel 4370  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-er 6129  df-ec 6131  df-qs 6135

This theorem is referenced by: (None)