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Theorem brresALTV 34368
Description: Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.)
Assertion
Ref Expression
brresALTV (𝐶𝑉 → (𝐵(𝑅𝐴)𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶)))

Proof of Theorem brresALTV
StepHypRef Expression
1 opelresALTV 34367 . 2 (𝐶𝑉 → (⟨𝐵, 𝐶⟩ ∈ (𝑅𝐴) ↔ (𝐵𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))
2 df-br 4787 . 2 (𝐵(𝑅𝐴)𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑅𝐴))
3 df-br 4787 . . 3 (𝐵𝑅𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝑅)
43anbi2i 609 . 2 ((𝐵𝐴𝐵𝑅𝐶) ↔ (𝐵𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅))
51, 2, 43bitr4g 303 1 (𝐶𝑉 → (𝐵(𝑅𝐴)𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wcel 2145  cop 4322   class class class wbr 4786  cres 5251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-res 5261
This theorem is referenced by:  brcnvepres  34369  brres2  34371  eldmres  34372  elecres  34378  brinxprnres  34395  exanres  34399  eqres  34444  alrmomorn  34458  alrmomodm  34459  brxrn  34471  rnxrnres  34492  1cossres  34519  eldm1cossres  34545  brssrres  34589
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