Theorem qliftel 6593

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 6591	. . 3 |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )
6	1, 5, 2	fliftel 5772	. 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 6558	. . . 4 |- ( ( ph /\ x e. X ) -> ( C R x <-> [ C ] R = [ x ] R ) )
10	9	anbi1d 462	. . 3 |- ( ( ph /\ x e. X ) -> ( ( C R x /\ D = A ) <-> ( [ C ] R = [ x ] R /\ D = A ) ) )
11	10	rexbidva 2467	. 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 1348   e. wcel 2141   E.wrex 2449   _Vcvv 2730   <.cop 3586   class class class wbr 3989   |-> cmpt 4050   ran crn 4612   Er wer 6510   [cec 6511   /.cqs 6512

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-er 6513  df-ec 6515  df-qs 6519

This theorem is referenced by: (None)