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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-imdirval2 | Structured version Visualization version GIF version |
Description: Value of the functionalized direct image. (Contributed by BJ, 16-Dec-2023.) |
Ref | Expression |
---|---|
bj-imdirval2.exa | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
bj-imdirval2.exb | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
bj-imdirval2.arg | ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵)) |
Ref | Expression |
---|---|
bj-imdirval2 | ⊢ (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-imdirval2.exa | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
2 | bj-imdirval2.exb | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
3 | 1, 2 | bj-imdirval 35348 | . 2 ⊢ (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑟 “ 𝑥) = 𝑦)})) |
4 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅) | |
5 | 4 | imaeq1d 5967 | . . . . 5 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (𝑟 “ 𝑥) = (𝑅 “ 𝑥)) |
6 | 5 | eqeq1d 2742 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → ((𝑟 “ 𝑥) = 𝑦 ↔ (𝑅 “ 𝑥) = 𝑦)) |
7 | 6 | anbi2d 629 | . . 3 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑟 “ 𝑥) = 𝑦) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦))) |
8 | 7 | opabbidv 5145 | . 2 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑟 “ 𝑥) = 𝑦)} = {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)}) |
9 | 1, 2 | xpexd 7595 | . . 3 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V) |
10 | bj-imdirval2.arg | . . 3 ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵)) | |
11 | 9, 10 | sselpwd 5254 | . 2 ⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵)) |
12 | 1, 2 | bj-imdirval2lem 35349 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)} ∈ V) |
13 | 3, 8, 11, 12 | fvmptd 6879 | 1 ⊢ (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 Vcvv 3431 ⊆ wss 3892 𝒫 cpw 4539 {copab 5141 × cxp 5588 “ cima 5593 ‘cfv 6432 (class class class)co 7271 𝒫*cimdir 35345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7274 df-oprab 7275 df-mpo 7276 df-imdir 35346 |
This theorem is referenced by: bj-imdirval3 35351 bj-imdirid 35353 bj-imdirco 35357 |
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