Theorem qliftel 6671

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 6669	. . 3 |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )
6	1, 5, 2	fliftel 5837	. 2 |- ( ph -> ( [ C ] R F D <-> E. x e. X ( [ C ] R = [ x ] R /\ D = A ) ) )
7	3	adantr 276	. . . . 5 |- ( ( ph /\ x e. X ) -> R Er X )
8		simpr 110	. . . . 5 |- ( ( ph /\ x e. X ) -> x e. X )
9	7, 8	erth2 6636	. . . 4 |- ( ( ph /\ x e. X ) -> ( C R x <-> [ C ] R = [ x ] R ) )
10	9	anbi1d 465	. . 3 |- ( ( ph /\ x e. X ) -> ( ( C R x /\ D = A ) <-> ( [ C ] R = [ x ] R /\ D = A ) ) )
11	10	rexbidva 2491	. 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 191	1 |- ( ph -> ( [ C ] R F D <-> E. x e. X ( C R x /\ D = A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   E.wrex 2473   _Vcvv 2760   <.cop 3622   class class class wbr 4030   |-> cmpt 4091   ran crn 4661   Er wer 6586   [cec 6587   /.cqs 6588

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-er 6589  df-ec 6591  df-qs 6595

This theorem is referenced by: (None)