Theorem fliftel 5453

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 3786	. . . 4 |- ( C F D <-> <. C , D >. e. F )
2		flift.1	. . . . 5 |- F = ran ( x e. X |-> <. A , B >. )
3	2	eleq2i 2145	. . . 4 |- ( <. C , D >. e. F <-> <. C , D >. e. ran ( x e. X |-> <. A , B >. ) )
4	1, 3	bitri 182	. . 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 3983	. . . . . 6 |- ( ( A e. R /\ B e. S ) -> <. A , B >. e. _V )
8	5, 6, 7	syl2anc 403	. . . . 5 |- ( ( ph /\ x e. X ) -> <. A , B >. e. _V )
9	8	ralrimiva 2434	. . . 4 |- ( ph -> A. x e. X <. A , B >. e. _V )
10		eqid 2081	. . . . 5 |- ( x e. X |-> <. A , B >. ) = ( x e. X |-> <. A , B >. )
11	10	elrnmptg 4604	. . . 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 190	. 2 |- ( ph -> ( C F D <-> E. x e. X <. C , D >. = <. A , B >. ) )
14		opthg2 3994	. . . 4 |- ( ( A e. R /\ B e. S ) -> ( <. C , D >. = <. A , B >. <-> ( C = A /\ D = B ) ) )
15	5, 6, 14	syl2anc 403	. . 3 |- ( ( ph /\ x e. X ) -> ( <. C , D >. = <. A , B >. <-> ( C = A /\ D = B ) ) )
16	15	rexbidva 2365	. 2 |- ( ph -> ( E. x e. X <. C , D >. = <. A , B >. <-> E. x e. X ( C = A /\ D = B ) ) )
17	13, 16	bitrd 186	1 |- ( ph -> ( C F D <-> E. x e. X ( C = A /\ D = B ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348   E.wrex 2349   _Vcvv 2601   <.cop 3401   class class class wbr 3785   |-> cmpt 3839   ran crn 4364

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-mpt 3841  df-cnv 4371  df-dm 4373  df-rn 4374

This theorem is referenced by:  fliftcnv  5455  fliftfun  5456  fliftf  5459  fliftval  5460  qliftel  6209