Theorem 1st2nd 7105

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 5045	. . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
2		ssel2 3563	. . 3 ⊢ ((𝐵 ⊆ (V × V) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (V × V))
3	1, 2	sylanb 488	. 2 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (V × V))
4		1st2nd2 7096	. 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 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ⟨cop 4131   × cxp 5036  Rel wrel 5043  ‘cfv 5804  1^st c1st 7057  2^nd c2nd 7058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-1st 7059  df-2nd 7060

This theorem is referenced by:  2ndrn  7107  1st2ndbr  7108  elopabi  7120  cnvf1olem  7162  ordpinq  9644  addassnq  9659  mulassnq  9660  distrnq  9662  mulidnq  9664  recmulnq  9665  ltexnq  9676  fsumcnv  14346  fprodcnv  14552  cofulid  16373  cofurid  16374  idffth  16416  cofull  16417  cofth  16418  ressffth  16421  isnat2  16431  nat1st2nd  16434  homadmcd  16515  catciso  16580  prf1st  16667  prf2nd  16668  1st2ndprf  16669  curfuncf  16701  uncfcurf  16702  curf2ndf  16710  yonffthlem  16745  yoniso  16748  dprd2dlem2  18262  dprd2dlem1  18263  dprd2da  18264  mdetunilem9  20245  2ndcctbss  21068  utop2nei  21864  utop3cls  21865  caubl  22914  elusuhgra  25897  wlkop  26056  nvop2  26847  nvvop  26848  nvop  26915  phop  27057  fgreu  28854  1stpreimas  28866  cvmliftlem1  30521  heiborlem3  32782  rngoi  32868  drngoi  32920  isdrngo1  32925  iscrngo2  32966  1wlkop  40832