![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-iminvval2 | Structured version Visualization version GIF version |
Description: Value of the functionalized inverse image. (Contributed by BJ, 23-May-2024.) |
Ref | Expression |
---|---|
bj-iminvval2.exa | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
bj-iminvval2.exb | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
bj-iminvval2.arg | ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵)) |
Ref | Expression |
---|---|
bj-iminvval2 | ⊢ (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑅 “ 𝑦))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-iminvval2.exa | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
2 | bj-iminvval2.exb | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
3 | 1, 2 | bj-iminvval 36012 | . 2 ⊢ (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑟 “ 𝑦))})) |
4 | simpr 486 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅) | |
5 | 4 | cnveqd 5873 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → ◡𝑟 = ◡𝑅) |
6 | 5 | imaeq1d 6056 | . . . . 5 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (◡𝑟 “ 𝑦) = (◡𝑅 “ 𝑦)) |
7 | 6 | eqeq2d 2744 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (𝑥 = (◡𝑟 “ 𝑦) ↔ 𝑥 = (◡𝑅 “ 𝑦))) |
8 | 7 | anbi2d 630 | . . 3 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑟 “ 𝑦)) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑅 “ 𝑦)))) |
9 | 8 | opabbidv 5213 | . 2 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑟 “ 𝑦))} = {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑅 “ 𝑦))}) |
10 | 1, 2 | xpexd 7733 | . . 3 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V) |
11 | bj-iminvval2.arg | . . 3 ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵)) | |
12 | 10, 11 | sselpwd 5325 | . 2 ⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵)) |
13 | 1, 2 | bj-imdirval2lem 36001 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑅 “ 𝑦))} ∈ V) |
14 | 3, 9, 12, 13 | fvmptd 7001 | 1 ⊢ (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑅 “ 𝑦))}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ⊆ wss 3947 𝒫 cpw 4601 {copab 5209 × cxp 5673 ◡ccnv 5674 “ cima 5678 ‘cfv 6540 (class class class)co 7404 𝒫*ciminv 36010 |
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-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
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-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-iminv 36011 |
This theorem is referenced by: bj-iminvid 36014 |
Copyright terms: Public domain | W3C validator |