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Theorem cnvrescnv 6215
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 5697 . . 3 (𝑅𝐵) = (𝑅 ∩ (𝐵 × V))
21cnveqi 5885 . 2 (𝑅𝐵) = (𝑅 ∩ (𝐵 × V))
3 cnvin 6164 . 2 (𝑅 ∩ (𝐵 × V)) = (𝑅(𝐵 × V))
4 cnvcnv 6212 . . . 4 𝑅 = (𝑅 ∩ (V × V))
5 cnvxp 6177 . . . 4 (𝐵 × V) = (V × 𝐵)
64, 5ineq12i 4218 . . 3 (𝑅(𝐵 × V)) = ((𝑅 ∩ (V × V)) ∩ (V × 𝐵))
7 inass 4228 . . 3 ((𝑅 ∩ (V × V)) ∩ (V × 𝐵)) = (𝑅 ∩ ((V × V) ∩ (V × 𝐵)))
8 inxp 5842 . . . . 5 ((V × V) ∩ (V × 𝐵)) = ((V ∩ V) × (V ∩ 𝐵))
9 inv1 4398 . . . . . . 7 (V ∩ V) = V
109eqcomi 2746 . . . . . 6 V = (V ∩ V)
11 ssv 4008 . . . . . . . 8 𝐵 ⊆ V
12 ssid 4006 . . . . . . . 8 𝐵𝐵
1311, 12ssini 4240 . . . . . . 7 𝐵 ⊆ (V ∩ 𝐵)
14 inss2 4238 . . . . . . 7 (V ∩ 𝐵) ⊆ 𝐵
1513, 14eqssi 4000 . . . . . 6 𝐵 = (V ∩ 𝐵)
1610, 15xpeq12i 5713 . . . . 5 (V × 𝐵) = ((V ∩ V) × (V ∩ 𝐵))
178, 16eqtr4i 2768 . . . 4 ((V × V) ∩ (V × 𝐵)) = (V × 𝐵)
1817ineq2i 4217 . . 3 (𝑅 ∩ ((V × V) ∩ (V × 𝐵))) = (𝑅 ∩ (V × 𝐵))
196, 7, 183eqtri 2769 . 2 (𝑅(𝐵 × V)) = (𝑅 ∩ (V × 𝐵))
202, 3, 193eqtri 2769 1 (𝑅𝐵) = (𝑅 ∩ (V × 𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  Vcvv 3480  cin 3950   × cxp 5683  ccnv 5684  cres 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-res 5697
This theorem is referenced by:  fressupp  32697
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