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| Description: Two ways to express the corestriction of a class. (Contributed by BJ, 28-Dec-2023.) | 
| Ref | Expression | 
|---|---|
| cnvrescnv | ⊢ ◡(◡𝑅 ↾ 𝐵) = (𝑅 ∩ (V × 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-res 5697 | . . 3 ⊢ (◡𝑅 ↾ 𝐵) = (◡𝑅 ∩ (𝐵 × V)) | |
| 2 | 1 | cnveqi 5885 | . 2 ⊢ ◡(◡𝑅 ↾ 𝐵) = ◡(◡𝑅 ∩ (𝐵 × V)) | 
| 3 | cnvin 6164 | . 2 ⊢ ◡(◡𝑅 ∩ (𝐵 × V)) = (◡◡𝑅 ∩ ◡(𝐵 × V)) | |
| 4 | cnvcnv 6212 | . . . 4 ⊢ ◡◡𝑅 = (𝑅 ∩ (V × V)) | |
| 5 | cnvxp 6177 | . . . 4 ⊢ ◡(𝐵 × V) = (V × 𝐵) | |
| 6 | 4, 5 | ineq12i 4218 | . . 3 ⊢ (◡◡𝑅 ∩ ◡(𝐵 × V)) = ((𝑅 ∩ (V × V)) ∩ (V × 𝐵)) | 
| 7 | inass 4228 | . . 3 ⊢ ((𝑅 ∩ (V × V)) ∩ (V × 𝐵)) = (𝑅 ∩ ((V × V) ∩ (V × 𝐵))) | |
| 8 | inxp 5842 | . . . . 5 ⊢ ((V × V) ∩ (V × 𝐵)) = ((V ∩ V) × (V ∩ 𝐵)) | |
| 9 | inv1 4398 | . . . . . . 7 ⊢ (V ∩ V) = V | |
| 10 | 9 | eqcomi 2746 | . . . . . 6 ⊢ V = (V ∩ V) | 
| 11 | ssv 4008 | . . . . . . . 8 ⊢ 𝐵 ⊆ V | |
| 12 | ssid 4006 | . . . . . . . 8 ⊢ 𝐵 ⊆ 𝐵 | |
| 13 | 11, 12 | ssini 4240 | . . . . . . 7 ⊢ 𝐵 ⊆ (V ∩ 𝐵) | 
| 14 | inss2 4238 | . . . . . . 7 ⊢ (V ∩ 𝐵) ⊆ 𝐵 | |
| 15 | 13, 14 | eqssi 4000 | . . . . . 6 ⊢ 𝐵 = (V ∩ 𝐵) | 
| 16 | 10, 15 | xpeq12i 5713 | . . . . 5 ⊢ (V × 𝐵) = ((V ∩ V) × (V ∩ 𝐵)) | 
| 17 | 8, 16 | eqtr4i 2768 | . . . 4 ⊢ ((V × V) ∩ (V × 𝐵)) = (V × 𝐵) | 
| 18 | 17 | ineq2i 4217 | . . 3 ⊢ (𝑅 ∩ ((V × V) ∩ (V × 𝐵))) = (𝑅 ∩ (V × 𝐵)) | 
| 19 | 6, 7, 18 | 3eqtri 2769 | . 2 ⊢ (◡◡𝑅 ∩ ◡(𝐵 × V)) = (𝑅 ∩ (V × 𝐵)) | 
| 20 | 2, 3, 19 | 3eqtri 2769 | 1 ⊢ ◡(◡𝑅 ↾ 𝐵) = (𝑅 ∩ (V × 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 Vcvv 3480 ∩ cin 3950 × cxp 5683 ◡ccnv 5684 ↾ cres 5687 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-res 5697 | 
| This theorem is referenced by: fressupp 32697 | 
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