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Theorem cnvrescnv 6204
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 5694 . . 3 (𝑅𝐵) = (𝑅 ∩ (𝐵 × V))
21cnveqi 5881 . 2 (𝑅𝐵) = (𝑅 ∩ (𝐵 × V))
3 cnvin 6154 . 2 (𝑅 ∩ (𝐵 × V)) = (𝑅(𝐵 × V))
4 cnvcnv 6201 . . . 4 𝑅 = (𝑅 ∩ (V × V))
5 cnvxp 6166 . . . 4 (𝐵 × V) = (V × 𝐵)
64, 5ineq12i 4212 . . 3 (𝑅(𝐵 × V)) = ((𝑅 ∩ (V × V)) ∩ (V × 𝐵))
7 inass 4222 . . 3 ((𝑅 ∩ (V × V)) ∩ (V × 𝐵)) = (𝑅 ∩ ((V × V) ∩ (V × 𝐵)))
8 inxp 5838 . . . . 5 ((V × V) ∩ (V × 𝐵)) = ((V ∩ V) × (V ∩ 𝐵))
9 inv1 4398 . . . . . . 7 (V ∩ V) = V
109eqcomi 2737 . . . . . 6 V = (V ∩ V)
11 ssv 4006 . . . . . . . 8 𝐵 ⊆ V
12 ssid 4004 . . . . . . . 8 𝐵𝐵
1311, 12ssini 4234 . . . . . . 7 𝐵 ⊆ (V ∩ 𝐵)
14 inss2 4232 . . . . . . 7 (V ∩ 𝐵) ⊆ 𝐵
1513, 14eqssi 3998 . . . . . 6 𝐵 = (V ∩ 𝐵)
1610, 15xpeq12i 5710 . . . . 5 (V × 𝐵) = ((V ∩ V) × (V ∩ 𝐵))
178, 16eqtr4i 2759 . . . 4 ((V × V) ∩ (V × 𝐵)) = (V × 𝐵)
1817ineq2i 4211 . . 3 (𝑅 ∩ ((V × V) ∩ (V × 𝐵))) = (𝑅 ∩ (V × 𝐵))
196, 7, 183eqtri 2760 . 2 (𝑅(𝐵 × V)) = (𝑅 ∩ (V × 𝐵))
202, 3, 193eqtri 2760 1 (𝑅𝐵) = (𝑅 ∩ (V × 𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  Vcvv 3473  cin 3948   × cxp 5680  ccnv 5681  cres 5684
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-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
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-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-res 5694
This theorem is referenced by:  fressupp  32497
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