Theorem 1st2nd 7880

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 5596	. . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
2		ssel2 3916	. . 3 ⊢ ((𝐵 ⊆ (V × V) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (V × V))
3	1, 2	sylanb 581	. 2 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (V × V))
4		1st2nd2 7870	. 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 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ⊆ wss 3887  ⟨cop 4567   × cxp 5587  Rel wrel 5594  ‘cfv 6433  1^st c1st 7829  2^nd c2nd 7830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fv 6441  df-1st 7831  df-2nd 7832

This theorem is referenced by:  2ndrn  7882  1st2ndbr  7883  funfv1st2nd  7887  funelss  7888  elopabi  7902  cnvf1olem  7950  ordpinq  10699  addassnq  10714  mulassnq  10715  distrnq  10717  mulidnq  10719  recmulnq  10720  ltexnq  10731  fsumcnv  15485  fprodcnv  15693  cofulid  17605  cofurid  17606  idffth  17649  cofull  17650  cofth  17651  ressffth  17654  isnat2  17664  nat1st2nd  17667  homadmcd  17757  catciso  17826  prf1st  17921  prf2nd  17922  1st2ndprf  17923  curfuncf  17956  uncfcurf  17957  curf2ndf  17965  yonffthlem  18000  yoniso  18003  dprd2dlem2  19643  dprd2dlem1  19644  dprd2da  19645  mdetunilem9  21769  2ndcctbss  22606  utop2nei  23402  utop3cls  23403  caubl  24472  wlkop  27995  nvop2  28970  nvvop  28971  nvop  29038  phop  29180  fgreu  31009  1stpreimas  31038  gsumhashmul  31316  cvmliftlem1  33247  heiborlem3  35971  rngoi  36057  drngoi  36109  isdrngo1  36114  iscrngo2  36155