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Theorem opelcnvg 4438
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.
StepHypRef Expression
1 breq2 3738 . . 3 (x = A → (y𝑅xy𝑅A))
2 breq1 3737 . . 3 (y = B → (y𝑅AB𝑅A))
3 df-cnv 4276 . . 3 𝑅 = {⟨x, y⟩ ∣ y𝑅x}
41, 2, 3brabg 3976 . 2 ((A 𝐶 B 𝐷) → (A𝑅BB𝑅A))
5 df-br 3735 . 2 (A𝑅B ↔ ⟨A, B 𝑅)
6 df-br 3735 . 2 (B𝑅A ↔ ⟨B, A 𝑅)
74, 5, 63bitr3g 211 1 ((A 𝐶 B 𝐷) → (⟨A, B 𝑅 ↔ ⟨B, A 𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wcel 1370  cop 3349   class class class wbr 3734  ccnv 4267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-cnv 4276
This theorem is referenced by:  brcnvg  4439  opelcnv  4440  fvimacnv  5203  cnvf1olem  5764  brtposg  5787  xrlenlt  6679
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