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Theorem brcnvepres 37795
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 brres 5986 . 2 (𝐶𝑊 → (𝐵( E ↾ 𝐴)𝐶 ↔ (𝐵𝐴𝐵 E 𝐶)))
2 brcnvep 37793 . . 3 (𝐵𝑉 → (𝐵 E 𝐶𝐶𝐵))
32anbi2d 628 . 2 (𝐵𝑉 → ((𝐵𝐴𝐵 E 𝐶) ↔ (𝐵𝐴𝐶𝐵)))
41, 3sylan9bbr 509 1 ((𝐵𝑉𝐶𝑊) → (𝐵( E ↾ 𝐴)𝐶 ↔ (𝐵𝐴𝐶𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wcel 2098   class class class wbr 5143   E cep 5575  ccnv 5671  cres 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-eprel 5576  df-xp 5678  df-rel 5679  df-cnv 5680  df-res 5684
This theorem is referenced by:  dfeldisj3  38247  dfeldisj4  38248
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