Theorem 1st2nd 5048

Description: Reconstruction of a member of a relation in terms of its ordered pair components.

Assertion

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

Proof of Theorem 1st2nd

Step	Hyp	Ref	Expression
1		ssel2 2616	. . 3 |- ((B C_ (_V X. _V) /\ A e. B) -> A e. (_V X. _V))
2		df-rel 4001	. . 3 |- (Rel B <-> B C_ (_V X. _V))
3	1, 2	sylanb 498	. 2 |- ((Rel B /\ A e. B) -> A e. (_V X. _V))
4		elxp6 5041	. . 3 |- (A e. (_V X. _V) <-> (A = <.(1st` A), (2nd` A)>. /\ ((1st` A) e. _V /\ (2nd` A) e. _V)))
5	4	simplbi 349	. 2 |- (A e. (_V X. _V) -> A = <.(1st` A), (2nd` A)>.)
6	3, 5	syl 12	1 |- ((Rel B /\ A e. B) -> A = <.(1st` A), (2nd` A)>.)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   C_ wss 2593  <.cop 3046   X. cxp 3984  Rel wrel 3991  ` cfv 3998  1stc1st 5018  2ndc2nd 5019

This theorem is referenced by:  2ndrn 5050  elopabi 5059  eloprabi 5060  fparlem1 5081  fparlem2 5082  drngi 9493  nvop2 9559  nvvop 9560  nvop 9637  ipfval 9691  phop 9818  oprabopabf 10157  11st22nd 14348  eloi 14400  issubcat 15193  isringd 16097  isdivrng1 16109  iscring2 16146

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-1st 5020  df-2nd 5021

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