Theorem 1st2ndbr 6163

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 6160	. . 3 |- ( ( Rel B /\ A e. B ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
2		simpr 109	. . 3 |- ( ( Rel B /\ A e. B ) -> A e. B )
3	1, 2	eqeltrrd 2248	. 2 |- ( ( Rel B /\ A e. B ) -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. B )
4		df-br 3990	. 2 |- ( ( 1st ` A ) B ( 2nd ` A ) <-> <. ( 1st ` A ) , ( 2nd ` A ) >. e. B )
5	3, 4	sylibr 133	1 |- ( ( Rel B /\ A e. B ) -> ( 1st ` A ) B ( 2nd ` A ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   <.cop 3586   class class class wbr 3989   Rel wrel 4616   ` cfv 5198   1stc1st 6117   2ndc2nd 6118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fv 5206  df-1st 6119  df-2nd 6120

This theorem is referenced by: (None)