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Theorem brinxprnres 36049
Description: Binary relation on a restriction. (Contributed by Peter Mazsa, 2-Jan-2019.)
Assertion
Ref Expression
brinxprnres (𝐶𝑉 → (𝐵(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶)))

Proof of Theorem brinxprnres
StepHypRef Expression
1 brres2 36030 . 2 (𝐵(𝑅𝐴)𝐶𝐵(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝐶)
2 brres 5832 . 2 (𝐶𝑉 → (𝐵(𝑅𝐴)𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶)))
31, 2bitr3id 288 1 (𝐶𝑉 → (𝐵(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2114  cin 3842   class class class wbr 5030   × cxp 5523  ran crn 5526  cres 5527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-xp 5531  df-rel 5532  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537
This theorem is referenced by:  inxprnres  36050
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