Theorem 1st2ndbr 6352

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 6349	. . 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 2308	. 2 |- ( ( Rel B /\ A e. B ) -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. B )
4		df-br 4090	. 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 2201   <.cop 3673   class class class wbr 4089   Rel wrel 4732   ` cfv 5328   1stc1st 6306   2ndc2nd 6307

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-sbc 3031  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-iota 5288  df-fun 5330  df-fv 5336  df-1st 6308  df-2nd 6309

This theorem is referenced by: (None)