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Theorem brres 4541
Description: Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)
Hypothesis
Ref Expression
opelres.1 B V
Assertion
Ref Expression
brres (A(𝐶𝐷)B ↔ (A𝐶B A 𝐷))

Proof of Theorem brres
StepHypRef Expression
1 opelres.1 . . 3 B V
21opelres 4540 . 2 (⟨A, B (𝐶𝐷) ↔ (⟨A, B 𝐶 A 𝐷))
3 df-br 3735 . 2 (A(𝐶𝐷)B ↔ ⟨A, B (𝐶𝐷))
4 df-br 3735 . . 3 (A𝐶B ↔ ⟨A, B 𝐶)
54anbi1i 434 . 2 ((A𝐶B A 𝐷) ↔ (⟨A, B 𝐶 A 𝐷))
62, 3, 53bitr4i 201 1 (A(𝐶𝐷)B ↔ (A𝐶B A 𝐷))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   wcel 1370  Vcvv 2531  cop 3349   class class class wbr 3734  cres 4270
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-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  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-xp 4274  df-res 4280
This theorem is referenced by:  dfres2  4581  dfima2  4593  poirr2  4640  cores  4747  resco  4748  rnco  4750  fnres  4937  fvres  5119  nfunsn  5128  1stconst  5761  2ndconst  5762
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