Theorem qliftel 6581

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
qliftel	|- ( ph -> ( [ C ] R F D <-> E. x e. X ( C R x /\ D = A ) ) )

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

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

Proof of Theorem qliftel

Step	Hyp	Ref	Expression
1		qlift.1	. . 3 |- F = ran ( x e. X |-> <. [ x ] R , A >. )
2		qlift.2	. . . 4 |- ( ( ph /\ x e. X ) -> A e. Y )
3		qlift.3	. . . 4 |- ( ph -> R Er X )
4		qlift.4	. . . 4 |- ( ph -> X e. _V )
5	1, 2, 3, 4	qliftlem 6579	. . 3 |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )
6	1, 5, 2	fliftel 5761	. 2 |- ( ph -> ( [ C ] R F D <-> E. x e. X ( [ C ] R = [ x ] R /\ D = A ) ) )
7	3	adantr 274	. . . . 5 |- ( ( ph /\ x e. X ) -> R Er X )
8		simpr 109	. . . . 5 |- ( ( ph /\ x e. X ) -> x e. X )
9	7, 8	erth2 6546	. . . 4 |- ( ( ph /\ x e. X ) -> ( C R x <-> [ C ] R = [ x ] R ) )
10	9	anbi1d 461	. . 3 |- ( ( ph /\ x e. X ) -> ( ( C R x /\ D = A ) <-> ( [ C ] R = [ x ] R /\ D = A ) ) )
11	10	rexbidva 2463	. 2 |- ( ph -> ( E. x e. X ( C R x /\ D = A ) <-> E. x e. X ( [ C ] R = [ x ] R /\ D = A ) ) )
12	6, 11	bitr4d 190	1 |- ( ph -> ( [ C ] R F D <-> E. x e. X ( C R x /\ D = A ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   E.wrex 2445   _Vcvv 2726   <.cop 3579   class class class wbr 3982   |-> cmpt 4043   ran crn 4605   Er wer 6498   [cec 6499   /.cqs 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-er 6501  df-ec 6503  df-qs 6507

This theorem is referenced by: (None)