Theorem fliftel1 5434

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
fliftel1	|- ( ( ph /\ x e. X ) -> A F B )

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

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

Proof of Theorem fliftel1

Step	Hyp	Ref	Expression
1		flift.2	. . . . 5 |- ( ( ph /\ x e. X ) -> A e. R )
2		flift.3	. . . . 5 |- ( ( ph /\ x e. X ) -> B e. S )
3		opexg 3964	. . . . 5 |- ( ( A e. R /\ B e. S ) -> <. A , B >. e. _V )
4	1, 2, 3	syl2anc 391	. . . 4 |- ( ( ph /\ x e. X ) -> <. A , B >. e. _V )
5		eqid 2040	. . . . . 6 |- ( x e. X |-> <. A , B >. ) = ( x e. X |-> <. A , B >. )
6	5	elrnmpt1 4585	. . . . 5 |- ( ( x e. X /\ <. A , B >. e. _V ) -> <. A , B >. e. ran ( x e. X |-> <. A , B >. ) )
7	6	adantll 445	. . . 4 |- ( ( ( ph /\ x e. X ) /\ <. A , B >. e. _V ) -> <. A , B >. e. ran ( x e. X |-> <. A , B >. ) )
8	4, 7	mpdan 398	. . 3 |- ( ( ph /\ x e. X ) -> <. A , B >. e. ran ( x e. X |-> <. A , B >. ) )
9		flift.1	. . 3 |- F = ran ( x e. X |-> <. A , B >. )
10	8, 9	syl6eleqr 2131	. 2 |- ( ( ph /\ x e. X ) -> <. A , B >. e. F )
11		df-br 3765	. 2 |- ( A F B <-> <. A , B >. e. F )
12	10, 11	sylibr 137	1 |- ( ( ph /\ x e. X ) -> A F B )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   _Vcvv 2557   <.cop 3378   class class class wbr 3764   |-> cmpt 3818   ran crn 4346

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-mpt 3820  df-cnv 4353  df-dm 4355  df-rn 4356

This theorem is referenced by:  fliftfun  5436  qliftel1  6187