Theorem qliftel 6372

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 6370	. . 3 |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )
6	1, 5, 2	fliftel 5572	. 2 |- ( ph -> ( [ C ] R F D <-> E. x e. X ( [ C ] R = [ x ] R /\ D = A ) ) )
7	3	adantr 270	. . . . 5 |- ( ( ph /\ x e. X ) -> R Er X )
8		simpr 108	. . . . 5 |- ( ( ph /\ x e. X ) -> x e. X )
9	7, 8	erth2 6337	. . . 4 |- ( ( ph /\ x e. X ) -> ( C R x <-> [ C ] R = [ x ] R ) )
10	9	anbi1d 453	. . 3 |- ( ( ph /\ x e. X ) -> ( ( C R x /\ D = A ) <-> ( [ C ] R = [ x ] R /\ D = A ) ) )
11	10	rexbidva 2377	. 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 189	1 |- ( ph -> ( [ C ] R F D <-> E. x e. X ( C R x /\ D = A ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1289   e. wcel 1438   E.wrex 2360   _Vcvv 2619   <.cop 3449   class class class wbr 3845   |-> cmpt 3899   ran crn 4439   Er wer 6289   [cec 6290   /.cqs 6291

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-er 6292  df-ec 6294  df-qs 6298

This theorem is referenced by: (None)