Theorem fliftel 5702

Description: Elementhood in the relation F. (Contributed by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
flift.1	|- F = ran ( x e. X |-> <. A , B >. )
flift.2	|- ( ( ph /\ x e. X ) -> A e. R )
flift.3	|- ( ( ph /\ x e. X ) -> B e. S )

Assertion

Ref	Expression
fliftel	|- ( ph -> ( C F D <-> E. x e. X ( C = A /\ D = B ) ) )

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

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

Proof of Theorem fliftel

Step	Hyp	Ref	Expression
1		df-br 3938	. . . 4 |- ( C F D <-> <. C , D >. e. F )
2		flift.1	. . . . 5 |- F = ran ( x e. X |-> <. A , B >. )
3	2	eleq2i 2207	. . . 4 |- ( <. C , D >. e. F <-> <. C , D >. e. ran ( x e. X |-> <. A , B >. ) )
4	1, 3	bitri 183	. . 3 |- ( C F D <-> <. C , D >. e. ran ( x e. X |-> <. A , B >. ) )
5		flift.2	. . . . . 6 |- ( ( ph /\ x e. X ) -> A e. R )
6		flift.3	. . . . . 6 |- ( ( ph /\ x e. X ) -> B e. S )
7		opexg 4158	. . . . . 6 |- ( ( A e. R /\ B e. S ) -> <. A , B >. e. _V )
8	5, 6, 7	syl2anc 409	. . . . 5 |- ( ( ph /\ x e. X ) -> <. A , B >. e. _V )
9	8	ralrimiva 2508	. . . 4 |- ( ph -> A. x e. X <. A , B >. e. _V )
10		eqid 2140	. . . . 5 |- ( x e. X |-> <. A , B >. ) = ( x e. X |-> <. A , B >. )
11	10	elrnmptg 4799	. . . 4 |- ( A. x e. X <. A , B >. e. _V -> ( <. C , D >. e. ran ( x e. X |-> <. A , B >. ) <-> E. x e. X <. C , D >. = <. A , B >. ) )
12	9, 11	syl 14	. . 3 |- ( ph -> ( <. C , D >. e. ran ( x e. X |-> <. A , B >. ) <-> E. x e. X <. C , D >. = <. A , B >. ) )
13	4, 12	syl5bb 191	. 2 |- ( ph -> ( C F D <-> E. x e. X <. C , D >. = <. A , B >. ) )
14		opthg2 4169	. . . 4 |- ( ( A e. R /\ B e. S ) -> ( <. C , D >. = <. A , B >. <-> ( C = A /\ D = B ) ) )
15	5, 6, 14	syl2anc 409	. . 3 |- ( ( ph /\ x e. X ) -> ( <. C , D >. = <. A , B >. <-> ( C = A /\ D = B ) ) )
16	15	rexbidva 2435	. 2 |- ( ph -> ( E. x e. X <. C , D >. = <. A , B >. <-> E. x e. X ( C = A /\ D = B ) ) )
17	13, 16	bitrd 187	1 |- ( ph -> ( C F D <-> E. x e. X ( C = A /\ D = B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   A.wral 2417   E.wrex 2418   _Vcvv 2689   <.cop 3535   class class class wbr 3937   |-> cmpt 3997   ran crn 4548

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-mpt 3999  df-cnv 4555  df-dm 4557  df-rn 4558

This theorem is referenced by:  fliftcnv  5704  fliftfun  5705  fliftf  5708  fliftval  5709  qliftel  6517