![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5649 | . . 3 ⊢ (◡𝑅 ↾ 𝐵) = (◡𝑅 ∩ (𝐵 × V)) | |
2 | 1 | cnveqi 5834 | . 2 ⊢ ◡(◡𝑅 ↾ 𝐵) = ◡(◡𝑅 ∩ (𝐵 × V)) |
3 | cnvin 6101 | . 2 ⊢ ◡(◡𝑅 ∩ (𝐵 × V)) = (◡◡𝑅 ∩ ◡(𝐵 × V)) | |
4 | cnvcnv 6148 | . . . 4 ⊢ ◡◡𝑅 = (𝑅 ∩ (V × V)) | |
5 | cnvxp 6113 | . . . 4 ⊢ ◡(𝐵 × V) = (V × 𝐵) | |
6 | 4, 5 | ineq12i 4174 | . . 3 ⊢ (◡◡𝑅 ∩ ◡(𝐵 × V)) = ((𝑅 ∩ (V × V)) ∩ (V × 𝐵)) |
7 | inass 4183 | . . 3 ⊢ ((𝑅 ∩ (V × V)) ∩ (V × 𝐵)) = (𝑅 ∩ ((V × V) ∩ (V × 𝐵))) | |
8 | inxp 5792 | . . . . 5 ⊢ ((V × V) ∩ (V × 𝐵)) = ((V ∩ V) × (V ∩ 𝐵)) | |
9 | inv1 4358 | . . . . . . 7 ⊢ (V ∩ V) = V | |
10 | 9 | eqcomi 2742 | . . . . . 6 ⊢ V = (V ∩ V) |
11 | ssv 3972 | . . . . . . . 8 ⊢ 𝐵 ⊆ V | |
12 | ssid 3970 | . . . . . . . 8 ⊢ 𝐵 ⊆ 𝐵 | |
13 | 11, 12 | ssini 4195 | . . . . . . 7 ⊢ 𝐵 ⊆ (V ∩ 𝐵) |
14 | inss2 4193 | . . . . . . 7 ⊢ (V ∩ 𝐵) ⊆ 𝐵 | |
15 | 13, 14 | eqssi 3964 | . . . . . 6 ⊢ 𝐵 = (V ∩ 𝐵) |
16 | 10, 15 | xpeq12i 5665 | . . . . 5 ⊢ (V × 𝐵) = ((V ∩ V) × (V ∩ 𝐵)) |
17 | 8, 16 | eqtr4i 2764 | . . . 4 ⊢ ((V × V) ∩ (V × 𝐵)) = (V × 𝐵) |
18 | 17 | ineq2i 4173 | . . 3 ⊢ (𝑅 ∩ ((V × V) ∩ (V × 𝐵))) = (𝑅 ∩ (V × 𝐵)) |
19 | 6, 7, 18 | 3eqtri 2765 | . 2 ⊢ (◡◡𝑅 ∩ ◡(𝐵 × V)) = (𝑅 ∩ (V × 𝐵)) |
20 | 2, 3, 19 | 3eqtri 2765 | 1 ⊢ ◡(◡𝑅 ↾ 𝐵) = (𝑅 ∩ (V × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 Vcvv 3447 ∩ cin 3913 × cxp 5635 ◡ccnv 5636 ↾ cres 5639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-xp 5643 df-rel 5644 df-cnv 5645 df-res 5649 |
This theorem is referenced by: fressupp 31656 |
Copyright terms: Public domain | W3C validator |