Theorem 1st2ndbr 6203

Description: Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.)

Assertion

Ref	Expression
1st2ndbr	|- ( ( Rel B /\ A e. B ) -> ( 1st ` A ) B ( 2nd ` A ) )

Proof of Theorem 1st2ndbr

Step	Hyp	Ref	Expression
1		1st2nd 6200	. . 3 |- ( ( Rel B /\ A e. B ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
2		simpr 110	. . 3 |- ( ( Rel B /\ A e. B ) -> A e. B )
3	1, 2	eqeltrrd 2267	. 2 |- ( ( Rel B /\ A e. B ) -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. B )
4		df-br 4019	. 2 |- ( ( 1st ` A ) B ( 2nd ` A ) <-> <. ( 1st ` A ) , ( 2nd ` A ) >. e. B )
5	3, 4	sylibr 134	1 |- ( ( Rel B /\ A e. B ) -> ( 1st ` A ) B ( 2nd ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2160   <.cop 3610   class class class wbr 4018   Rel wrel 4646   ` cfv 5231   1stc1st 6157   2ndc2nd 6158

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-iota 5193  df-fun 5233  df-fv 5239  df-1st 6159  df-2nd 6160

This theorem is referenced by: (None)