Theorem cnvrescnv 6098

Description: Two ways to express the corestriction of a class. (Contributed by BJ, 28-Dec-2023.)

Assertion

Ref	Expression
cnvrescnv	⊢ ◡(◡𝑅 ↾ 𝐵) = (𝑅 ∩ (V × 𝐵))

Proof of Theorem cnvrescnv

Step	Hyp	Ref	Expression
1		df-res 5601	. . 3 ⊢ (◡𝑅 ↾ 𝐵) = (◡𝑅 ∩ (𝐵 × V))
2	1	cnveqi 5783	. 2 ⊢ ◡(◡𝑅 ↾ 𝐵) = ◡(◡𝑅 ∩ (𝐵 × V))
3		cnvin 6048	. 2 ⊢ ◡(◡𝑅 ∩ (𝐵 × V)) = (◡◡𝑅 ∩ ◡(𝐵 × V))
4		cnvcnv 6095	. . . 4 ⊢ ◡◡𝑅 = (𝑅 ∩ (V × V))
5		cnvxp 6060	. . . 4 ⊢ ◡(𝐵 × V) = (V × 𝐵)
6	4, 5	ineq12i 4144	. . 3 ⊢ (◡◡𝑅 ∩ ◡(𝐵 × V)) = ((𝑅 ∩ (V × V)) ∩ (V × 𝐵))
7		inass 4153	. . 3 ⊢ ((𝑅 ∩ (V × V)) ∩ (V × 𝐵)) = (𝑅 ∩ ((V × V) ∩ (V × 𝐵)))
8		inxp 5741	. . . . 5 ⊢ ((V × V) ∩ (V × 𝐵)) = ((V ∩ V) × (V ∩ 𝐵))
9		inv1 4328	. . . . . . 7 ⊢ (V ∩ V) = V
10	9	eqcomi 2747	. . . . . 6 ⊢ V = (V ∩ V)
11		ssv 3945	. . . . . . . 8 ⊢ 𝐵 ⊆ V
12		ssid 3943	. . . . . . . 8 ⊢ 𝐵 ⊆ 𝐵
13	11, 12	ssini 4165	. . . . . . 7 ⊢ 𝐵 ⊆ (V ∩ 𝐵)
14		inss2 4163	. . . . . . 7 ⊢ (V ∩ 𝐵) ⊆ 𝐵
15	13, 14	eqssi 3937	. . . . . 6 ⊢ 𝐵 = (V ∩ 𝐵)
16	10, 15	xpeq12i 5617	. . . . 5 ⊢ (V × 𝐵) = ((V ∩ V) × (V ∩ 𝐵))
17	8, 16	eqtr4i 2769	. . . 4 ⊢ ((V × V) ∩ (V × 𝐵)) = (V × 𝐵)
18	17	ineq2i 4143	. . 3 ⊢ (𝑅 ∩ ((V × V) ∩ (V × 𝐵))) = (𝑅 ∩ (V × 𝐵))
19	6, 7, 18	3eqtri 2770	. 2 ⊢ (◡◡𝑅 ∩ ◡(𝐵 × V)) = (𝑅 ∩ (V × 𝐵))
20	2, 3, 19	3eqtri 2770	1 ⊢ ◡(◡𝑅 ↾ 𝐵) = (𝑅 ∩ (V × 𝐵))

Syntax hints:   = wceq 1539  Vcvv 3432   ∩ cin 3886   × cxp 5587  ^◡ccnv 5588   ↾ cres 5591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-res 5601

This theorem is referenced by:  fressupp  31022