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Theorem cnvrescnv 6028
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 5543 . . 3 (𝑅𝐵) = (𝑅 ∩ (𝐵 × V))
21cnveqi 5721 . 2 (𝑅𝐵) = (𝑅 ∩ (𝐵 × V))
3 cnvin 5979 . 2 (𝑅 ∩ (𝐵 × V)) = (𝑅(𝐵 × V))
4 cnvcnv 6025 . . . 4 𝑅 = (𝑅 ∩ (V × V))
5 cnvxp 5990 . . . 4 (𝐵 × V) = (V × 𝐵)
64, 5ineq12i 4165 . . 3 (𝑅(𝐵 × V)) = ((𝑅 ∩ (V × V)) ∩ (V × 𝐵))
7 inass 4174 . . 3 ((𝑅 ∩ (V × V)) ∩ (V × 𝐵)) = (𝑅 ∩ ((V × V) ∩ (V × 𝐵)))
8 inxp 5679 . . . . 5 ((V × V) ∩ (V × 𝐵)) = ((V ∩ V) × (V ∩ 𝐵))
9 inv1 4324 . . . . . . 7 (V ∩ V) = V
109eqcomi 2829 . . . . . 6 V = (V ∩ V)
11 ssv 3970 . . . . . . . 8 𝐵 ⊆ V
12 ssid 3968 . . . . . . . 8 𝐵𝐵
1311, 12ssini 4186 . . . . . . 7 𝐵 ⊆ (V ∩ 𝐵)
14 inss2 4184 . . . . . . 7 (V ∩ 𝐵) ⊆ 𝐵
1513, 14eqssi 3962 . . . . . 6 𝐵 = (V ∩ 𝐵)
1610, 15xpeq12i 5559 . . . . 5 (V × 𝐵) = ((V ∩ V) × (V ∩ 𝐵))
178, 16eqtr4i 2846 . . . 4 ((V × V) ∩ (V × 𝐵)) = (V × 𝐵)
1817ineq2i 4164 . . 3 (𝑅 ∩ ((V × V) ∩ (V × 𝐵))) = (𝑅 ∩ (V × 𝐵))
196, 7, 183eqtri 2847 . 2 (𝑅(𝐵 × V)) = (𝑅 ∩ (V × 𝐵))
202, 3, 193eqtri 2847 1 (𝑅𝐵) = (𝑅 ∩ (V × 𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  Vcvv 3473  cin 3912   × cxp 5529  ccnv 5530  cres 5533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-br 5043  df-opab 5105  df-xp 5537  df-rel 5538  df-cnv 5539  df-res 5543
This theorem is referenced by: (None)
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