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Theorem brres2 35516
Description: Binary relation on a restriction. (Contributed by Peter Mazsa, 2-Jan-2019.) (Revised by Peter Mazsa, 16-Dec-2021.)
Assertion
Ref Expression
brres2 (𝐵(𝑅𝐴)𝐶𝐵(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝐶)

Proof of Theorem brres2
StepHypRef Expression
1 brres 5853 . . 3 (𝐶 ∈ ran (𝑅𝐴) → (𝐵(𝑅𝐴)𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶)))
21pm5.32i 577 . 2 ((𝐶 ∈ ran (𝑅𝐴) ∧ 𝐵(𝑅𝐴)𝐶) ↔ (𝐶 ∈ ran (𝑅𝐴) ∧ (𝐵𝐴𝐵𝑅𝐶)))
3 relres 5875 . . . 4 Rel (𝑅𝐴)
43relelrni 5812 . . 3 (𝐵(𝑅𝐴)𝐶𝐶 ∈ ran (𝑅𝐴))
54pm4.71ri 563 . 2 (𝐵(𝑅𝐴)𝐶 ↔ (𝐶 ∈ ran (𝑅𝐴) ∧ 𝐵(𝑅𝐴)𝐶))
6 brinxp2 5622 . . 3 (𝐵(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝐶 ↔ ((𝐵𝐴𝐶 ∈ ran (𝑅𝐴)) ∧ 𝐵𝑅𝐶))
7 df-3an 1083 . . 3 ((𝐵𝐴𝐶 ∈ ran (𝑅𝐴) ∧ 𝐵𝑅𝐶) ↔ ((𝐵𝐴𝐶 ∈ ran (𝑅𝐴)) ∧ 𝐵𝑅𝐶))
8 3anan12 1090 . . 3 ((𝐵𝐴𝐶 ∈ ran (𝑅𝐴) ∧ 𝐵𝑅𝐶) ↔ (𝐶 ∈ ran (𝑅𝐴) ∧ (𝐵𝐴𝐵𝑅𝐶)))
96, 7, 83bitr2i 301 . 2 (𝐵(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝐶 ↔ (𝐶 ∈ ran (𝑅𝐴) ∧ (𝐵𝐴𝐵𝑅𝐶)))
102, 5, 93bitr4i 305 1 (𝐵(𝑅𝐴)𝐶𝐵(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝐶)
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398  w3a 1081  wcel 2107  cin 3933   class class class wbr 5057   × cxp 5546  ran crn 5549  cres 5550
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560
This theorem is referenced by:  brinxprnres  35535
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