Theorem qliftel 6617

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 6615	. . 3 |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )
6	1, 5, 2	fliftel 5796	. 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 6582	. . . 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 2474	. 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 1353   e. wcel 2148   E.wrex 2456   _Vcvv 2739   <.cop 3597   class class class wbr 4005   |-> cmpt 4066   ran crn 4629   Er wer 6534   [cec 6535   /.cqs 6536

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-er 6537  df-ec 6539  df-qs 6543

This theorem is referenced by: (None)