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Theorem 1st2ndbr 6848
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
StepHypRef Expression
1 1st2nd 6845 . . 3  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2 simpr 461 . . 3  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  e.  B )
31, 2eqeltrrd 2546 . 2  |-  ( ( Rel  B  /\  A  e.  B )  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  B )
4 df-br 4457 . 2  |-  ( ( 1st `  A ) B ( 2nd `  A
)  <->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  B
)
53, 4sylibr 212 1  |-  ( ( Rel  B  /\  A  e.  B )  ->  ( 1st `  A ) B ( 2nd `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1819   <.cop 4038   class class class wbr 4456   Rel wrel 5013   ` cfv 5594   1stc1st 6797   2ndc2nd 6798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fv 5602  df-1st 6799  df-2nd 6800
This theorem is referenced by:  cofuval  15298  cofu1  15300  cofu2  15302  cofucl  15304  cofuass  15305  cofulid  15306  cofurid  15307  funcres  15312  cofull  15350  cofth  15351  isnat2  15364  fuccocl  15380  fucidcl  15381  fuclid  15382  fucrid  15383  fucass  15384  fucsect  15388  fucinv  15389  invfuc  15390  fuciso  15391  natpropd  15392  fucpropd  15393  homahom  15445  homadm  15446  homacd  15447  homadmcd  15448  catciso  15513  prfval  15595  prfcl  15599  prf1st  15600  prf2nd  15601  1st2ndprf  15602  evlfcllem  15617  evlfcl  15618  curf1cl  15624  curf2cl  15627  curfcl  15628  uncf1  15632  uncf2  15633  curfuncf  15634  uncfcurf  15635  diag1cl  15638  diag2cl  15642  curf2ndf  15643  yon1cl  15659  oyon1cl  15667  yonedalem1  15668  yonedalem21  15669  yonedalem3a  15670  yonedalem4c  15673  yonedalem22  15674  yonedalem3b  15675  yonedalem3  15676  yonedainv  15677  yonffthlem  15678  yoniso  15681  utop2nei  20879  utop3cls  20880
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