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Theorem cnvrescnv 6168

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 5650	. . 3 ⊢ (^◡𝑅 ↾ 𝐵) = (^◡𝑅 ∩ (𝐵 × V))
2	1	cnveqi 5838	. 2 ⊢ ^◡(^◡𝑅 ↾ 𝐵) = ^◡(^◡𝑅 ∩ (𝐵 × V))
3		cnvin 6117	. 2 ⊢ ^◡(^◡𝑅 ∩ (𝐵 × V)) = (^◡^◡𝑅 ∩ ^◡(𝐵 × V))
4		cnvcnv 6165	. . . 4 ⊢ ^◡^◡𝑅 = (𝑅 ∩ (V × V))
5		cnvxp 6130	. . . 4 ⊢ ^◡(𝐵 × V) = (V × 𝐵)
6	4, 5	ineq12i 4181	. . 3 ⊢ (^◡^◡𝑅 ∩ ^◡(𝐵 × V)) = ((𝑅 ∩ (V × V)) ∩ (V × 𝐵))
7		inass 4191	. . 3 ⊢ ((𝑅 ∩ (V × V)) ∩ (V × 𝐵)) = (𝑅 ∩ ((V × V) ∩ (V × 𝐵)))
8		inxp 5795	. . . . 5 ⊢ ((V × V) ∩ (V × 𝐵)) = ((V ∩ V) × (V ∩ 𝐵))
9		inv1 4361	. . . . . . 7 ⊢ (V ∩ V) = V
10	9	eqcomi 2738	. . . . . 6 ⊢ V = (V ∩ V)
11		ssv 3971	. . . . . . . 8 ⊢ 𝐵 ⊆ V
12		ssid 3969	. . . . . . . 8 ⊢ 𝐵 ⊆ 𝐵
13	11, 12	ssini 4203	. . . . . . 7 ⊢ 𝐵 ⊆ (V ∩ 𝐵)
14		inss2 4201	. . . . . . 7 ⊢ (V ∩ 𝐵) ⊆ 𝐵
15	13, 14	eqssi 3963	. . . . . 6 ⊢ 𝐵 = (V ∩ 𝐵)
16	10, 15	xpeq12i 5666	. . . . 5 ⊢ (V × 𝐵) = ((V ∩ V) × (V ∩ 𝐵))
17	8, 16	eqtr4i 2755	. . . 4 ⊢ ((V × V) ∩ (V × 𝐵)) = (V × 𝐵)
18	17	ineq2i 4180	. . 3 ⊢ (𝑅 ∩ ((V × V) ∩ (V × 𝐵))) = (𝑅 ∩ (V × 𝐵))
19	6, 7, 18	3eqtri 2756	. 2 ⊢ (^◡^◡𝑅 ∩ ^◡(𝐵 × V)) = (𝑅 ∩ (V × 𝐵))
20	2, 3, 19	3eqtri 2756	1 ⊢ ^◡(^◡𝑅 ↾ 𝐵) = (𝑅 ∩ (V × 𝐵))

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 Vcvv 3447 ∩ cin 3913 × cxp 5636 ^◡ccnv 5637 ↾ cres 5640

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-11 2158 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pr 5387

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2721 df-clel 2803 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-sn 4590 df-pr 4592 df-op 4596 df-br 5108 df-opab 5170 df-xp 5644 df-rel 5645 df-cnv 5646 df-res 5650

This theorem is referenced by: fressupp 32611