Theorem fliftel 6559

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 4654	. . 3 |- ( C F D <-> <. C , D >. e. F )
2		flift.1	. . . 4 |- F = ran ( x e. X |-> <. A , B >. )
3	2	eleq2i 2693	. . 3 |- ( <. C , D >. e. F <-> <. C , D >. e. ran ( x e. X |-> <. A , B >. ) )
4		eqid 2622	. . . 4 |- ( x e. X |-> <. A , B >. ) = ( x e. X |-> <. A , B >. )
5		opex 4932	. . . 4 |- <. A , B >. e. _V
6	4, 5	elrnmpti 5376	. . 3 |- ( <. C , D >. e. ran ( x e. X |-> <. A , B >. ) <-> E. x e. X <. C , D >. = <. A , B >. )
7	1, 3, 6	3bitri 286	. 2 |- ( C F D <-> E. x e. X <. C , D >. = <. A , B >. )
8		flift.2	. . . 4 |- ( ( ph /\ x e. X ) -> A e. R )
9		flift.3	. . . 4 |- ( ( ph /\ x e. X ) -> B e. S )
10		opthg2 4948	. . . 4 |- ( ( A e. R /\ B e. S ) -> ( <. C , D >. = <. A , B >. <-> ( C = A /\ D = B ) ) )
11	8, 9, 10	syl2anc 693	. . 3 |- ( ( ph /\ x e. X ) -> ( <. C , D >. = <. A , B >. <-> ( C = A /\ D = B ) ) )
12	11	rexbidva 3049	. 2 |- ( ph -> ( E. x e. X <. C , D >. = <. A , B >. <-> E. x e. X ( C = A /\ D = B ) ) )
13	7, 12	syl5bb 272	1 |- ( ph -> ( C F D <-> E. x e. X ( C = A /\ D = B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   ran crn 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-mpt 4730  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  fliftcnv  6561  fliftfun  6562  fliftf  6565  fliftval  6566  qliftel  7830