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Theorem cnvrescnv 6087

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 5592	. . 3 ⊢ (^◡𝑅 ↾ 𝐵) = (^◡𝑅 ∩ (𝐵 × V))
2	1	cnveqi 5772	. 2 ⊢ ^◡(^◡𝑅 ↾ 𝐵) = ^◡(^◡𝑅 ∩ (𝐵 × V))
3		cnvin 6037	. 2 ⊢ ^◡(^◡𝑅 ∩ (𝐵 × V)) = (^◡^◡𝑅 ∩ ^◡(𝐵 × V))
4		cnvcnv 6084	. . . 4 ⊢ ^◡^◡𝑅 = (𝑅 ∩ (V × V))
5		cnvxp 6049	. . . 4 ⊢ ^◡(𝐵 × V) = (V × 𝐵)
6	4, 5	ineq12i 4141	. . 3 ⊢ (^◡^◡𝑅 ∩ ^◡(𝐵 × V)) = ((𝑅 ∩ (V × V)) ∩ (V × 𝐵))
7		inass 4150	. . 3 ⊢ ((𝑅 ∩ (V × V)) ∩ (V × 𝐵)) = (𝑅 ∩ ((V × V) ∩ (V × 𝐵)))
8		inxp 5730	. . . . 5 ⊢ ((V × V) ∩ (V × 𝐵)) = ((V ∩ V) × (V ∩ 𝐵))
9		inv1 4325	. . . . . . 7 ⊢ (V ∩ V) = V
10	9	eqcomi 2747	. . . . . 6 ⊢ V = (V ∩ V)
11		ssv 3941	. . . . . . . 8 ⊢ 𝐵 ⊆ V
12		ssid 3939	. . . . . . . 8 ⊢ 𝐵 ⊆ 𝐵
13	11, 12	ssini 4162	. . . . . . 7 ⊢ 𝐵 ⊆ (V ∩ 𝐵)
14		inss2 4160	. . . . . . 7 ⊢ (V ∩ 𝐵) ⊆ 𝐵
15	13, 14	eqssi 3933	. . . . . 6 ⊢ 𝐵 = (V ∩ 𝐵)
16	10, 15	xpeq12i 5608	. . . . 5 ⊢ (V × 𝐵) = ((V ∩ V) × (V ∩ 𝐵))
17	8, 16	eqtr4i 2769	. . . 4 ⊢ ((V × V) ∩ (V × 𝐵)) = (V × 𝐵)
18	17	ineq2i 4140	. . 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))

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 Vcvv 3422 ∩ cin 3882 × cxp 5578 ^◡ccnv 5579 ↾ cres 5582

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2817 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-br 5071 df-opab 5133 df-xp 5586 df-rel 5587 df-cnv 5588 df-res 5592

This theorem is referenced by: fressupp 30924