Theorem 1st2ndbr 6242

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 6239	. . 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 2274	. 2 |- ( ( Rel B /\ A e. B ) -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. B )
4		df-br 4034	. 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 2167   <.cop 3625   class class class wbr 4033   Rel wrel 4668   ` cfv 5258   1stc1st 6196   2ndc2nd 6197

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fv 5266  df-1st 6198  df-2nd 6199

This theorem is referenced by: (None)