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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 35099 | . 2 ⊢ (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑟 “ 𝑦))})) |
4 | simpr 488 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅) | |
5 | 4 | cnveqd 5744 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → ◡𝑟 = ◡𝑅) |
6 | 5 | imaeq1d 5928 | . . . . 5 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (◡𝑟 “ 𝑦) = (◡𝑅 “ 𝑦)) |
7 | 6 | eqeq2d 2748 | . . . 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 5119 | . 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 7536 | . . 3 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V) |
11 | bj-iminvval2.arg | . . 3 ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵)) | |
12 | 10, 11 | sselpwd 5219 | . 2 ⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵)) |
13 | 1, 2 | bj-imdirval2lem 35088 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑅 “ 𝑦))} ∈ V) |
14 | 3, 9, 12, 13 | fvmptd 6825 | 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 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 ⊆ wss 3866 𝒫 cpw 4513 {copab 5115 × cxp 5549 ◡ccnv 5550 “ cima 5554 ‘cfv 6380 (class class class)co 7213 𝒫*ciminv 35097 |
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 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
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 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-iminv 35098 |
This theorem is referenced by: bj-iminvid 35101 |
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