Theorem 1st2nd 6388

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 4761	. . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
2		ssel2 3237	. . 3 ⊢ ((𝐵 ⊆ (V × V) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (V × V))
3	1, 2	sylanb 284	. 2 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (V × V))
4		1st2nd2 6382	. 2 ⊢ (𝐴 ∈ (V × V) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
5	3, 4	syl 14	1 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  Vcvv 2815   ⊆ wss 3214  ⟨cop 3697   × cxp 4752  Rel wrel 4759  ‘cfv 5357  1^st c1st 6345  2^nd c2nd 6346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fv 5365  df-1st 6347  df-2nd 6348

This theorem is referenced by:  2ndrn  6390  1st2ndbr  6391  elopabi  6404  cnvf1olem  6433  elmpom  6447  exmidapne  7590  aptap  8941  fsumcnv  12148  fprodcnv  12336  wlkop  16469