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Theorem 1st2ndbr 6830
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 6827 . . 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 2556 . 2  |-  ( ( Rel  B  /\  A  e.  B )  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  B )
4 df-br 4448 . 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 1767   <.cop 4033   class class class wbr 4447   Rel wrel 5004   ` cfv 5586   1stc1st 6779   2ndc2nd 6780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fv 5594  df-1st 6781  df-2nd 6782
This theorem is referenced by:  cofuval  15105  cofu1  15107  cofu2  15109  cofucl  15111  cofuass  15112  cofulid  15113  cofurid  15114  funcres  15119  cofull  15157  cofth  15158  isnat2  15171  fuccocl  15187  fucidcl  15188  fuclid  15189  fucrid  15190  fucass  15191  fucsect  15195  fucinv  15196  invfuc  15197  fuciso  15198  natpropd  15199  fucpropd  15200  homahom  15220  homadm  15221  homacd  15222  homadmcd  15223  catciso  15288  prfval  15322  prfcl  15326  prf1st  15327  prf2nd  15328  1st2ndprf  15329  evlfcllem  15344  evlfcl  15345  curf1cl  15351  curf2cl  15354  curfcl  15355  uncf1  15359  uncf2  15360  curfuncf  15361  uncfcurf  15362  diag1cl  15365  diag2cl  15369  curf2ndf  15370  yon1cl  15386  oyon1cl  15394  yonedalem1  15395  yonedalem21  15396  yonedalem3a  15397  yonedalem4c  15400  yonedalem22  15401  yonedalem3b  15402  yonedalem3  15403  yonedainv  15404  yonffthlem  15405  yoniso  15408  utop2nei  20488  utop3cls  20489
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