Theorem 1st2nd 7732

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 ∈ 𝐵) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)

Proof of Theorem 1st2nd

Step	Hyp	Ref	Expression
1		df-rel 5557	. . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
2		ssel2 3962	. . 3 ⊢ ((𝐵 ⊆ (V × V) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (V × V))
3	1, 2	sylanb 583	. 2 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (V × V))
4		1st2nd2 7722	. 2 ⊢ (𝐴 ∈ (V × V) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
5	3, 4	syl 17	1 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Vcvv 3495   ⊆ wss 3936  ⟨cop 4567   × cxp 5548  Rel wrel 5555  ‘cfv 6350  1^st c1st 7681  2^nd c2nd 7682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-iota 6309  df-fun 6352  df-fv 6358  df-1st 7683  df-2nd 7684

This theorem is referenced by:  2ndrn  7734  1st2ndbr  7735  funfv1st2nd  7739  funelss  7740  elopabi  7754  cnvf1olem  7799  ordpinq  10359  addassnq  10374  mulassnq  10375  distrnq  10377  mulidnq  10379  recmulnq  10380  ltexnq  10391  fsumcnv  15122  fprodcnv  15331  cofulid  17154  cofurid  17155  idffth  17197  cofull  17198  cofth  17199  ressffth  17202  isnat2  17212  nat1st2nd  17215  homadmcd  17296  catciso  17361  prf1st  17448  prf2nd  17449  1st2ndprf  17450  curfuncf  17482  uncfcurf  17483  curf2ndf  17491  yonffthlem  17526  yoniso  17529  dprd2dlem2  19156  dprd2dlem1  19157  dprd2da  19158  mdetunilem9  21223  2ndcctbss  22057  utop2nei  22853  utop3cls  22854  caubl  23905  wlkop  27403  nvop2  28379  nvvop  28380  nvop  28447  phop  28589  fgreu  30411  1stpreimas  30435  cvmliftlem1  32527  heiborlem3  35085  rngoi  35171  drngoi  35223  isdrngo1  35228  iscrngo2  35269