Theorem 1st2nd 6158

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 4616	. . 3 |- ( Rel B <-> B C_ ( _V X. _V ) )
2		ssel2 3142	. . 3 |- ( ( B C_ ( _V X. _V ) /\ A e. B ) -> A e. ( _V X. _V ) )
3	1, 2	sylanb 282	. 2 |- ( ( Rel B /\ A e. B ) -> A e. ( _V X. _V ) )
4		1st2nd2 6152	. 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 103   = wceq 1348   e. wcel 2141   _Vcvv 2730   C_ wss 3121   <.cop 3584   X. cxp 4607   Rel wrel 4614   ` cfv 5196   1stc1st 6115   2ndc2nd 6116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-iota 5158  df-fun 5198  df-fv 5204  df-1st 6117  df-2nd 6118

This theorem is referenced by:  2ndrn  6160  1st2ndbr  6161  elopabi  6172  cnvf1olem  6201  fsumcnv  11389  fprodcnv  11577