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Theorem cnvrescnv 6194

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 5673	. . 3 ⊢ (^◡𝑅 ↾ 𝐵) = (^◡𝑅 ∩ (𝐵 × V))
2	1	cnveqi 5860	. 2 ⊢ ^◡(^◡𝑅 ↾ 𝐵) = ^◡(^◡𝑅 ∩ (𝐵 × V))
3		cnvin 6141	. 2 ⊢ ^◡(^◡𝑅 ∩ (𝐵 × V)) = (^◡^◡𝑅 ∩ ^◡(𝐵 × V))
4		cnvcnv 6190	. . . 4 ⊢ ^◡^◡𝑅 = (𝑅 ∩ (V × V))
5		cnvxp 6154	. . . 4 ⊢ ^◡(𝐵 × V) = (V × 𝐵)
6	4, 5	ineq12i 4170	. . 3 ⊢ (^◡^◡𝑅 ∩ ^◡(𝐵 × V)) = ((𝑅 ∩ (V × V)) ∩ (V × 𝐵))
7		inass 4179	. . 3 ⊢ ((𝑅 ∩ (V × V)) ∩ (V × 𝐵)) = (𝑅 ∩ ((V × V) ∩ (V × 𝐵)))
8		inxp 5818	. . . . 5 ⊢ ((V × V) ∩ (V × 𝐵)) = ((V ∩ V) × (V ∩ 𝐵))
9		inv1 4354	. . . . . . 7 ⊢ (V ∩ V) = V
10	9	eqcomi 2770	. . . . . 6 ⊢ V = (V ∩ V)
11		ssv 3960	. . . . . . . 8 ⊢ 𝐵 ⊆ V
12		ssid 3958	. . . . . . . 8 ⊢ 𝐵 ⊆ 𝐵
13	11, 12	ssini 4191	. . . . . . 7 ⊢ 𝐵 ⊆ (V ∩ 𝐵)
14		inss2 4189	. . . . . . 7 ⊢ (V ∩ 𝐵) ⊆ 𝐵
15	13, 14	eqssi 3952	. . . . . 6 ⊢ 𝐵 = (V ∩ 𝐵)
16	10, 15	xpeq12i 5689	. . . . 5 ⊢ (V × 𝐵) = ((V ∩ V) × (V ∩ 𝐵))
17	8, 16	eqtr4i 2787	. . . 4 ⊢ ((V × V) ∩ (V × 𝐵)) = (V × 𝐵)
18	17	ineq2i 4169	. . 3 ⊢ (𝑅 ∩ ((V × V) ∩ (V × 𝐵))) = (𝑅 ∩ (V × 𝐵))
19	6, 7, 18	3eqtri 2788	. 2 ⊢ (^◡^◡𝑅 ∩ ^◡(𝐵 × V)) = (𝑅 ∩ (V × 𝐵))
20	2, 3, 19	3eqtri 2788	1 ⊢ ^◡(^◡𝑅 ↾ 𝐵) = (𝑅 ∩ (V × 𝐵))

Colors of variables: wff setvar class

Syntax hints: = wceq 1568 Vcvv 3453 ∩ cin 3903 × cxp 5659 ^◡ccnv 5660 ↾ cres 5663

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-11 2190 ax-ext 2733 ax-sep 5256 ax-pr 5404

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-sb 2095 df-clab 2740 df-cleq 2753 df-clel 2836 df-ral 3078 df-rex 3088 df-rab 3415 df-v 3455 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-sn 4589 df-pr 4591 df-op 4595 df-br 5109 df-opab 5173 df-xp 5667 df-rel 5668 df-cnv 5669 df-res 5673

This theorem is referenced by: fressupp 32999