Theorem fliftel1 5945

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 4326	. . . . 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 2231	. . . . . 6 |- ( x e. X |-> <. A , B >. ) = ( x e. X |-> <. A , B >. )
6	5	elrnmpt1 4989	. . . . 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 2325	. 2 |- ( ( ph /\ x e. X ) -> <. A , B >. e. F )
11		df-br 4094	. 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 1398   e. wcel 2202   _Vcvv 2803   <.cop 3676   class class class wbr 4093   |-> cmpt 4155   ran crn 4732

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-mpt 4157  df-cnv 4739  df-dm 4741  df-rn 4742

This theorem is referenced by:  fliftfun  5947  qliftel1  6828