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Theorem brcnvepres 34353
Description: Restricted converse epsilon binary relation. (Contributed by Peter Mazsa, 10-Feb-2018.)
Assertion
Ref Expression
brcnvepres ((𝐵𝑉𝐶𝑊) → (𝐵( E ↾ 𝐴)𝐶 ↔ (𝐵𝐴𝐶𝐵)))

Proof of Theorem brcnvepres
StepHypRef Expression
1 brresALTV 34352 . 2 (𝐶𝑊 → (𝐵( E ↾ 𝐴)𝐶 ↔ (𝐵𝐴𝐵 E 𝐶)))
2 brcnvep 34349 . . 3 (𝐵𝑉 → (𝐵 E 𝐶𝐶𝐵))
32anbi2d 742 . 2 (𝐵𝑉 → ((𝐵𝐴𝐵 E 𝐶) ↔ (𝐵𝐴𝐶𝐵)))
41, 3sylan9bbr 739 1 ((𝐵𝑉𝐶𝑊) → (𝐵( E ↾ 𝐴)𝐶 ↔ (𝐵𝐴𝐶𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2135   class class class wbr 4800   E cep 5174  ccnv 5261  cres 5264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pr 5051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4801  df-opab 4861  df-eprel 5175  df-xp 5268  df-rel 5269  df-cnv 5270  df-res 5274
This theorem is referenced by: (None)
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