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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 34873 | . 2 ⊢ (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑟 “ 𝑦))})) |
4 | simpr 489 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅) | |
5 | 4 | cnveqd 5708 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → ◡𝑟 = ◡𝑅) |
6 | 5 | imaeq1d 5893 | . . . . 5 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (◡𝑟 “ 𝑦) = (◡𝑅 “ 𝑦)) |
7 | 6 | eqeq2d 2770 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (𝑥 = (◡𝑟 “ 𝑦) ↔ 𝑥 = (◡𝑅 “ 𝑦))) |
8 | 7 | anbi2d 632 | . . 3 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑟 “ 𝑦)) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑅 “ 𝑦)))) |
9 | 8 | opabbidv 5091 | . 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 7465 | . . 3 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V) |
11 | bj-iminvval2.arg | . . 3 ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵)) | |
12 | 10, 11 | sselpwd 5189 | . 2 ⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵)) |
13 | 1, 2 | bj-imdirval2lem 34862 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑅 “ 𝑦))} ∈ V) |
14 | 3, 9, 12, 13 | fvmptd 6759 | 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 400 = wceq 1539 ∈ wcel 2112 Vcvv 3407 ⊆ wss 3854 𝒫 cpw 4487 {copab 5087 × cxp 5515 ◡ccnv 5516 “ cima 5520 ‘cfv 6328 (class class class)co 7143 𝒫*ciminv 34871 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-ov 7146 df-oprab 7147 df-mpo 7148 df-iminv 34872 |
This theorem is referenced by: bj-iminvid 34875 |
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