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Theorem cnvrescnv 6217

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 5701	. . 3 ⊢ (^◡𝑅 ↾ 𝐵) = (^◡𝑅 ∩ (𝐵 × V))
2	1	cnveqi 5888	. 2 ⊢ ^◡(^◡𝑅 ↾ 𝐵) = ^◡(^◡𝑅 ∩ (𝐵 × V))
3		cnvin 6167	. 2 ⊢ ^◡(^◡𝑅 ∩ (𝐵 × V)) = (^◡^◡𝑅 ∩ ^◡(𝐵 × V))
4		cnvcnv 6214	. . . 4 ⊢ ^◡^◡𝑅 = (𝑅 ∩ (V × V))
5		cnvxp 6179	. . . 4 ⊢ ^◡(𝐵 × V) = (V × 𝐵)
6	4, 5	ineq12i 4226	. . 3 ⊢ (^◡^◡𝑅 ∩ ^◡(𝐵 × V)) = ((𝑅 ∩ (V × V)) ∩ (V × 𝐵))
7		inass 4236	. . 3 ⊢ ((𝑅 ∩ (V × V)) ∩ (V × 𝐵)) = (𝑅 ∩ ((V × V) ∩ (V × 𝐵)))
8		inxp 5845	. . . . 5 ⊢ ((V × V) ∩ (V × 𝐵)) = ((V ∩ V) × (V ∩ 𝐵))
9		inv1 4404	. . . . . . 7 ⊢ (V ∩ V) = V
10	9	eqcomi 2744	. . . . . 6 ⊢ V = (V ∩ V)
11		ssv 4020	. . . . . . . 8 ⊢ 𝐵 ⊆ V
12		ssid 4018	. . . . . . . 8 ⊢ 𝐵 ⊆ 𝐵
13	11, 12	ssini 4248	. . . . . . 7 ⊢ 𝐵 ⊆ (V ∩ 𝐵)
14		inss2 4246	. . . . . . 7 ⊢ (V ∩ 𝐵) ⊆ 𝐵
15	13, 14	eqssi 4012	. . . . . 6 ⊢ 𝐵 = (V ∩ 𝐵)
16	10, 15	xpeq12i 5717	. . . . 5 ⊢ (V × 𝐵) = ((V ∩ V) × (V ∩ 𝐵))
17	8, 16	eqtr4i 2766	. . . 4 ⊢ ((V × V) ∩ (V × 𝐵)) = (V × 𝐵)
18	17	ineq2i 4225	. . 3 ⊢ (𝑅 ∩ ((V × V) ∩ (V × 𝐵))) = (𝑅 ∩ (V × 𝐵))
19	6, 7, 18	3eqtri 2767	. 2 ⊢ (^◡^◡𝑅 ∩ ^◡(𝐵 × V)) = (𝑅 ∩ (V × 𝐵))
20	2, 3, 19	3eqtri 2767	1 ⊢ ^◡(^◡𝑅 ↾ 𝐵) = (𝑅 ∩ (V × 𝐵))

Colors of variables: wff setvar class

Syntax hints: = wceq 1537 Vcvv 3478 ∩ cin 3962 × cxp 5687 ^◡ccnv 5688 ↾ cres 5691

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-res 5701

This theorem is referenced by: fressupp 32703