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Theorem cnvrescnv 6148
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 5631 . . 3 (𝑅𝐵) = (𝑅 ∩ (𝐵 × V))
21cnveqi 5819 . 2 (𝑅𝐵) = (𝑅 ∩ (𝐵 × V))
3 cnvin 6097 . 2 (𝑅 ∩ (𝐵 × V)) = (𝑅(𝐵 × V))
4 cnvcnv 6145 . . . 4 𝑅 = (𝑅 ∩ (V × V))
5 cnvxp 6110 . . . 4 (𝐵 × V) = (V × 𝐵)
64, 5ineq12i 4167 . . 3 (𝑅(𝐵 × V)) = ((𝑅 ∩ (V × V)) ∩ (V × 𝐵))
7 inass 4177 . . 3 ((𝑅 ∩ (V × V)) ∩ (V × 𝐵)) = (𝑅 ∩ ((V × V) ∩ (V × 𝐵)))
8 inxp 5776 . . . . 5 ((V × V) ∩ (V × 𝐵)) = ((V ∩ V) × (V ∩ 𝐵))
9 inv1 4347 . . . . . . 7 (V ∩ V) = V
109eqcomi 2740 . . . . . 6 V = (V ∩ V)
11 ssv 3954 . . . . . . . 8 𝐵 ⊆ V
12 ssid 3952 . . . . . . . 8 𝐵𝐵
1311, 12ssini 4189 . . . . . . 7 𝐵 ⊆ (V ∩ 𝐵)
14 inss2 4187 . . . . . . 7 (V ∩ 𝐵) ⊆ 𝐵
1513, 14eqssi 3946 . . . . . 6 𝐵 = (V ∩ 𝐵)
1610, 15xpeq12i 5647 . . . . 5 (V × 𝐵) = ((V ∩ V) × (V ∩ 𝐵))
178, 16eqtr4i 2757 . . . 4 ((V × V) ∩ (V × 𝐵)) = (V × 𝐵)
1817ineq2i 4166 . . 3 (𝑅 ∩ ((V × V) ∩ (V × 𝐵))) = (𝑅 ∩ (V × 𝐵))
196, 7, 183eqtri 2758 . 2 (𝑅(𝐵 × V)) = (𝑅 ∩ (V × 𝐵))
202, 3, 193eqtri 2758 1 (𝑅𝐵) = (𝑅 ∩ (V × 𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  Vcvv 3436  cin 3896   × cxp 5617  ccnv 5618  cres 5621
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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-11 2160  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-xp 5625  df-rel 5626  df-cnv 5627  df-res 5631
This theorem is referenced by:  fressupp  32676
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