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Theorem cnvrescnv 6184
Description: Two ways to express the corestriction of a class. (Contributed by BJ, 28-Dec-2023.)
Assertion
Ref Expression
cnvrescnv (𝑅𝐵) = (𝑅 ∩ (V × 𝐵))

Proof of Theorem cnvrescnv
StepHypRef Expression
1 df-res 5682 . . 3 (𝑅𝐵) = (𝑅 ∩ (𝐵 × V))
21cnveqi 5867 . 2 (𝑅𝐵) = (𝑅 ∩ (𝐵 × V))
3 cnvin 6134 . 2 (𝑅 ∩ (𝐵 × V)) = (𝑅(𝐵 × V))
4 cnvcnv 6181 . . . 4 𝑅 = (𝑅 ∩ (V × V))
5 cnvxp 6146 . . . 4 (𝐵 × V) = (V × 𝐵)
64, 5ineq12i 4207 . . 3 (𝑅(𝐵 × V)) = ((𝑅 ∩ (V × V)) ∩ (V × 𝐵))
7 inass 4216 . . 3 ((𝑅 ∩ (V × V)) ∩ (V × 𝐵)) = (𝑅 ∩ ((V × V) ∩ (V × 𝐵)))
8 inxp 5825 . . . . 5 ((V × V) ∩ (V × 𝐵)) = ((V ∩ V) × (V ∩ 𝐵))
9 inv1 4391 . . . . . . 7 (V ∩ V) = V
109eqcomi 2741 . . . . . 6 V = (V ∩ V)
11 ssv 4003 . . . . . . . 8 𝐵 ⊆ V
12 ssid 4001 . . . . . . . 8 𝐵𝐵
1311, 12ssini 4228 . . . . . . 7 𝐵 ⊆ (V ∩ 𝐵)
14 inss2 4226 . . . . . . 7 (V ∩ 𝐵) ⊆ 𝐵
1513, 14eqssi 3995 . . . . . 6 𝐵 = (V ∩ 𝐵)
1610, 15xpeq12i 5698 . . . . 5 (V × 𝐵) = ((V ∩ V) × (V ∩ 𝐵))
178, 16eqtr4i 2763 . . . 4 ((V × V) ∩ (V × 𝐵)) = (V × 𝐵)
1817ineq2i 4206 . . 3 (𝑅 ∩ ((V × V) ∩ (V × 𝐵))) = (𝑅 ∩ (V × 𝐵))
196, 7, 183eqtri 2764 . 2 (𝑅(𝐵 × V)) = (𝑅 ∩ (V × 𝐵))
202, 3, 193eqtri 2764 1 (𝑅𝐵) = (𝑅 ∩ (V × 𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  Vcvv 3474  cin 3944   × cxp 5668  ccnv 5669  cres 5672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5143  df-opab 5205  df-xp 5676  df-rel 5677  df-cnv 5678  df-res 5682
This theorem is referenced by:  fressupp  31845
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