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Theorem cnvrescnv 6188

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 5681	. . 3 ⊢ (^◡𝑅 ↾ 𝐵) = (^◡𝑅 ∩ (𝐵 × V))
2	1	cnveqi 5868	. 2 ⊢ ^◡(^◡𝑅 ↾ 𝐵) = ^◡(^◡𝑅 ∩ (𝐵 × V))
3		cnvin 6138	. 2 ⊢ ^◡(^◡𝑅 ∩ (𝐵 × V)) = (^◡^◡𝑅 ∩ ^◡(𝐵 × V))
4		cnvcnv 6185	. . . 4 ⊢ ^◡^◡𝑅 = (𝑅 ∩ (V × V))
5		cnvxp 6150	. . . 4 ⊢ ^◡(𝐵 × V) = (V × 𝐵)
6	4, 5	ineq12i 4205	. . 3 ⊢ (^◡^◡𝑅 ∩ ^◡(𝐵 × V)) = ((𝑅 ∩ (V × V)) ∩ (V × 𝐵))
7		inass 4214	. . 3 ⊢ ((𝑅 ∩ (V × V)) ∩ (V × 𝐵)) = (𝑅 ∩ ((V × V) ∩ (V × 𝐵)))
8		inxp 5825	. . . . 5 ⊢ ((V × V) ∩ (V × 𝐵)) = ((V ∩ V) × (V ∩ 𝐵))
9		inv1 4389	. . . . . . 7 ⊢ (V ∩ V) = V
10	9	eqcomi 2735	. . . . . 6 ⊢ V = (V ∩ V)
11		ssv 4001	. . . . . . . 8 ⊢ 𝐵 ⊆ V
12		ssid 3999	. . . . . . . 8 ⊢ 𝐵 ⊆ 𝐵
13	11, 12	ssini 4226	. . . . . . 7 ⊢ 𝐵 ⊆ (V ∩ 𝐵)
14		inss2 4224	. . . . . . 7 ⊢ (V ∩ 𝐵) ⊆ 𝐵
15	13, 14	eqssi 3993	. . . . . 6 ⊢ 𝐵 = (V ∩ 𝐵)
16	10, 15	xpeq12i 5697	. . . . 5 ⊢ (V × 𝐵) = ((V ∩ V) × (V ∩ 𝐵))
17	8, 16	eqtr4i 2757	. . . 4 ⊢ ((V × V) ∩ (V × 𝐵)) = (V × 𝐵)
18	17	ineq2i 4204	. . 3 ⊢ (𝑅 ∩ ((V × V) ∩ (V × 𝐵))) = (𝑅 ∩ (V × 𝐵))
19	6, 7, 18	3eqtri 2758	. 2 ⊢ (^◡^◡𝑅 ∩ ^◡(𝐵 × V)) = (𝑅 ∩ (V × 𝐵))
20	2, 3, 19	3eqtri 2758	1 ⊢ ^◡(^◡𝑅 ↾ 𝐵) = (𝑅 ∩ (V × 𝐵))

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 Vcvv 3468 ∩ cin 3942 × cxp 5667 ^◡ccnv 5668 ↾ cres 5671

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 df-cnv 5677 df-res 5681

This theorem is referenced by: fressupp 32417