Theorem qliftel1 6464

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 6461	. 2 |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )
6	1, 5, 2	fliftel1 5649	1 |- ( ( ph /\ x e. X ) -> [ x ] R F A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1314   e. wcel 1463   _Vcvv 2657   <.cop 3496   class class class wbr 3895   |-> cmpt 3949   ran crn 4500   Er wer 6380   [cec 6381   /.cqs 6382

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-sbc 2879  df-csb 2972  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-mpt 3951  df-xp 4505  df-rel 4506  df-cnv 4507  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-er 6383  df-ec 6385  df-qs 6389

This theorem is referenced by: (None)