Theorem fliftel 5885

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 4060	. . . 4 |- ( C F D <-> <. C , D >. e. F )
2		flift.1	. . . . 5 |- F = ran ( x e. X |-> <. A , B >. )
3	2	eleq2i 2274	. . . 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 4290	. . . . . 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 2581	. . . 4 |- ( ph -> A. x e. X <. A , B >. e. _V )
10		eqid 2207	. . . . 5 |- ( x e. X |-> <. A , B >. ) = ( x e. X |-> <. A , B >. )
11	10	elrnmptg 4949	. . . 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 4301	. . . 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 2505	. 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 1373   e. wcel 2178   A.wral 2486   E.wrex 2487   _Vcvv 2776   <.cop 3646   class class class wbr 4059   |-> cmpt 4121   ran crn 4694

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-mpt 4123  df-cnv 4701  df-dm 4703  df-rn 4704

This theorem is referenced by:  fliftcnv  5887  fliftfun  5888  fliftf  5891  fliftval  5892  qliftel  6725