Theorem fliftel 5933

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 4089	. . . 4 |- ( C F D <-> <. C , D >. e. F )
2		flift.1	. . . . 5 |- F = ran ( x e. X |-> <. A , B >. )
3	2	eleq2i 2298	. . . 4 |- ( <. C , D >. e. F <-> <. C , D >. e. ran ( x e. X |-> <. A , B >. ) )
4	1, 3	bitri 184	. . 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 4320	. . . . . 6 |- ( ( A e. R /\ B e. S ) -> <. A , B >. e. _V )
8	5, 6, 7	syl2anc 411	. . . . 5 |- ( ( ph /\ x e. X ) -> <. A , B >. e. _V )
9	8	ralrimiva 2605	. . . 4 |- ( ph -> A. x e. X <. A , B >. e. _V )
10		eqid 2231	. . . . 5 |- ( x e. X |-> <. A , B >. ) = ( x e. X |-> <. A , B >. )
11	10	elrnmptg 4984	. . . 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	bitrid 192	. 2 |- ( ph -> ( C F D <-> E. x e. X <. C , D >. = <. A , B >. ) )
14		opthg2 4331	. . . 4 |- ( ( A e. R /\ B e. S ) -> ( <. C , D >. = <. A , B >. <-> ( C = A /\ D = B ) ) )
15	5, 6, 14	syl2anc 411	. . 3 |- ( ( ph /\ x e. X ) -> ( <. C , D >. = <. A , B >. <-> ( C = A /\ D = B ) ) )
16	15	rexbidva 2529	. 2 |- ( ph -> ( E. x e. X <. C , D >. = <. A , B >. <-> E. x e. X ( C = A /\ D = B ) ) )
17	13, 16	bitrd 188	1 |- ( ph -> ( C F D <-> E. x e. X ( C = A /\ D = B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   A.wral 2510   E.wrex 2511   _Vcvv 2802   <.cop 3672   class class class wbr 4088   |-> cmpt 4150   ran crn 4726

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-mpt 4152  df-cnv 4733  df-dm 4735  df-rn 4736

This theorem is referenced by:  fliftcnv  5935  fliftfun  5936  fliftf  5939  fliftval  5940  qliftel  6783