Theorem fliftel1 5863

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 4272	. . . . 5 |- ( ( A e. R /\ B e. S ) -> <. A , B >. e. _V )
4	1, 2, 3	syl2anc 411	. . . 4 |- ( ( ph /\ x e. X ) -> <. A , B >. e. _V )
5		eqid 2205	. . . . . 6 |- ( x e. X |-> <. A , B >. ) = ( x e. X |-> <. A , B >. )
6	5	elrnmpt1 4929	. . . . 5 |- ( ( x e. X /\ <. A , B >. e. _V ) -> <. A , B >. e. ran ( x e. X |-> <. A , B >. ) )
7	6	adantll 476	. . . 4 |- ( ( ( ph /\ x e. X ) /\ <. A , B >. e. _V ) -> <. A , B >. e. ran ( x e. X |-> <. A , B >. ) )
8	4, 7	mpdan 421	. . 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	eleqtrrdi 2299	. 2 |- ( ( ph /\ x e. X ) -> <. A , B >. e. F )
11		df-br 4045	. 2 |- ( A F B <-> <. A , B >. e. F )
12	10, 11	sylibr 134	1 |- ( ( ph /\ x e. X ) -> A F B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   _Vcvv 2772   <.cop 3636   class class class wbr 4044   |-> cmpt 4105   ran crn 4676

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-mpt 4107  df-cnv 4683  df-dm 4685  df-rn 4686

This theorem is referenced by:  fliftfun  5865  qliftel1  6703