Theorem qliftel1 6884

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 6881	. 2 |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )
6	1, 5, 2	fliftel1 5994	1 |- ( ( ph /\ x e. X ) -> [ x ] R F A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   <.cop 3711   class class class wbr 4128   |-> cmpt 4190   ran crn 4773   Er wer 6798   [cec 6799   /.cqs 6800

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-xp 4778  df-rel 4779  df-cnv 4780  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-er 6801  df-ec 6803  df-qs 6807

This theorem is referenced by: (None)