Theorem 1st2nd 6266

Description: Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)

Assertion

Ref	Expression
1st2nd	|- ( ( Rel B /\ A e. B ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )

Proof of Theorem 1st2nd

Step	Hyp	Ref	Expression
1		df-rel 4681	. . 3 |- ( Rel B <-> B C_ ( _V X. _V ) )
2		ssel2 3187	. . 3 |- ( ( B C_ ( _V X. _V ) /\ A e. B ) -> A e. ( _V X. _V ) )
3	1, 2	sylanb 284	. 2 |- ( ( Rel B /\ A e. B ) -> A e. ( _V X. _V ) )
4		1st2nd2 6260	. 2 |- ( A e. ( _V X. _V ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
5	3, 4	syl 14	1 |- ( ( Rel B /\ A e. B ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   _Vcvv 2771   C_ wss 3165   <.cop 3635   X. cxp 4672   Rel wrel 4679   ` cfv 5270   1stc1st 6223   2ndc2nd 6224

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-iota 5231  df-fun 5272  df-fv 5278  df-1st 6225  df-2nd 6226

This theorem is referenced by:  2ndrn  6268  1st2ndbr  6269  elopabi  6280  cnvf1olem  6309  exmidapne  7371  aptap  8722  fsumcnv  11719  fprodcnv  11907