Theorem 1st2nd 6408

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 4779	. . 3 |- ( Rel B <-> B C_ ( _V X. _V ) )
2		ssel2 3243	. . 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 6402	. 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 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   <.cop 3711   X. cxp 4770   Rel wrel 4777   ` cfv 5375   1stc1st 6365   2ndc2nd 6366

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-iota 5335  df-fun 5377  df-fv 5383  df-1st 6367  df-2nd 6368

This theorem is referenced by:  2ndrn  6410  1st2ndbr  6411  elopabi  6424  cnvf1olem  6453  elmpom  6467  exmidapne  7619  aptap  8971  fsumcnv  12185  fprodcnv  12373  wlkop  16506