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Mirrors > Home > MPE Home > Th. List > cnvrescnv | Structured version Visualization version GIF version |
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 5688 | . . 3 ⊢ (◡𝑅 ↾ 𝐵) = (◡𝑅 ∩ (𝐵 × V)) | |
2 | 1 | cnveqi 5874 | . 2 ⊢ ◡(◡𝑅 ↾ 𝐵) = ◡(◡𝑅 ∩ (𝐵 × V)) |
3 | cnvin 6144 | . 2 ⊢ ◡(◡𝑅 ∩ (𝐵 × V)) = (◡◡𝑅 ∩ ◡(𝐵 × V)) | |
4 | cnvcnv 6191 | . . . 4 ⊢ ◡◡𝑅 = (𝑅 ∩ (V × V)) | |
5 | cnvxp 6156 | . . . 4 ⊢ ◡(𝐵 × V) = (V × 𝐵) | |
6 | 4, 5 | ineq12i 4210 | . . 3 ⊢ (◡◡𝑅 ∩ ◡(𝐵 × V)) = ((𝑅 ∩ (V × V)) ∩ (V × 𝐵)) |
7 | inass 4219 | . . 3 ⊢ ((𝑅 ∩ (V × V)) ∩ (V × 𝐵)) = (𝑅 ∩ ((V × V) ∩ (V × 𝐵))) | |
8 | inxp 5832 | . . . . 5 ⊢ ((V × V) ∩ (V × 𝐵)) = ((V ∩ V) × (V ∩ 𝐵)) | |
9 | inv1 4394 | . . . . . . 7 ⊢ (V ∩ V) = V | |
10 | 9 | eqcomi 2741 | . . . . . 6 ⊢ V = (V ∩ V) |
11 | ssv 4006 | . . . . . . . 8 ⊢ 𝐵 ⊆ V | |
12 | ssid 4004 | . . . . . . . 8 ⊢ 𝐵 ⊆ 𝐵 | |
13 | 11, 12 | ssini 4231 | . . . . . . 7 ⊢ 𝐵 ⊆ (V ∩ 𝐵) |
14 | inss2 4229 | . . . . . . 7 ⊢ (V ∩ 𝐵) ⊆ 𝐵 | |
15 | 13, 14 | eqssi 3998 | . . . . . 6 ⊢ 𝐵 = (V ∩ 𝐵) |
16 | 10, 15 | xpeq12i 5704 | . . . . 5 ⊢ (V × 𝐵) = ((V ∩ V) × (V ∩ 𝐵)) |
17 | 8, 16 | eqtr4i 2763 | . . . 4 ⊢ ((V × V) ∩ (V × 𝐵)) = (V × 𝐵) |
18 | 17 | ineq2i 4209 | . . 3 ⊢ (𝑅 ∩ ((V × V) ∩ (V × 𝐵))) = (𝑅 ∩ (V × 𝐵)) |
19 | 6, 7, 18 | 3eqtri 2764 | . 2 ⊢ (◡◡𝑅 ∩ ◡(𝐵 × V)) = (𝑅 ∩ (V × 𝐵)) |
20 | 2, 3, 19 | 3eqtri 2764 | 1 ⊢ ◡(◡𝑅 ↾ 𝐵) = (𝑅 ∩ (V × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 Vcvv 3474 ∩ cin 3947 × cxp 5674 ◡ccnv 5675 ↾ cres 5678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 df-res 5688 |
This theorem is referenced by: fressupp 31905 |
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