Theorem opelcnvg 4727

Description: Ordered-pair membership in converse. (Contributed by NM, 13-May-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
opelcnvg	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⟨𝐴, 𝐵⟩ ∈ ◡𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))

Proof of Theorem opelcnvg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 3941	. . 3 ⊢ (𝑥 = 𝐴 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐴))
2		breq1 3940	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐴 ↔ 𝐵𝑅𝐴))
3		df-cnv 4555	. . 3 ⊢ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥}
4	1, 2, 3	brabg 4199	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))
5		df-br 3938	. 2 ⊢ (𝐴◡𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ◡𝑅)
6		df-br 3938	. 2 ⊢ (𝐵𝑅𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅)
7	4, 5, 6	3bitr3g 221	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⟨𝐴, 𝐵⟩ ∈ ◡𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∈ wcel 1481  ⟨cop 3535   class class class wbr 3937  ^◡ccnv 4546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-cnv 4555

This theorem is referenced by:  brcnvg  4728  opelcnv  4729  fvimacnv  5543  cnvf1olem  6129  brtposg  6159  xrlenlt  7853