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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-iminvval | Structured version Visualization version GIF version |
Description: Value of the functionalized inverse image. (Contributed by BJ, 23-May-2024.) |
Ref | Expression |
---|---|
bj-iminvval.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
bj-iminvval.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
Ref | Expression |
---|---|
bj-iminvval | ⊢ (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑟 “ 𝑦))})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-iminvval.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
2 | bj-iminvval.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
3 | df-iminv 35033 | . 2 ⊢ 𝒫* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝑥 = (◡𝑟 “ 𝑦))})) | |
4 | 1, 2, 3 | bj-imdirvallem 35021 | 1 ⊢ (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑟 “ 𝑦))})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2115 ⊆ wss 3857 𝒫 cpw 4503 {copab 5105 ↦ cmpt 5124 × cxp 5538 ◡ccnv 5539 “ cima 5543 (class class class)co 7195 𝒫*ciminv 35032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2021 ax-8 2117 ax-9 2125 ax-10 2146 ax-11 2163 ax-12 2180 ax-ext 2712 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7505 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2076 df-mo 2542 df-eu 2572 df-clab 2719 df-cleq 2732 df-clel 2813 df-nfc 2883 df-ne 2937 df-ral 3060 df-rex 3061 df-reu 3062 df-rab 3064 df-v 3403 df-sbc 3687 df-csb 3802 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6320 df-fun 6364 df-fn 6365 df-f 6366 df-f1 6367 df-fo 6368 df-f1o 6369 df-fv 6370 df-ov 7198 df-oprab 7199 df-mpo 7200 df-iminv 35033 |
This theorem is referenced by: bj-iminvval2 35035 |
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