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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 34475 | . 2 ⊢ (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑟 “ 𝑥) = 𝑦)})) |
4 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅) | |
5 | 4 | imaeq1d 5928 | . . . . 5 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (𝑟 “ 𝑥) = (𝑅 “ 𝑥)) |
6 | 5 | eqeq1d 2823 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → ((𝑟 “ 𝑥) = 𝑦 ↔ (𝑅 “ 𝑥) = 𝑦)) |
7 | 6 | anbi2d 630 | . . 3 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑟 “ 𝑥) = 𝑦) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦))) |
8 | 7 | opabbidv 5132 | . 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 7474 | . . 3 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V) |
10 | bj-imdirval2.arg | . . 3 ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵)) | |
11 | 9, 10 | sselpwd 5230 | . 2 ⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵)) |
12 | 1 | pwexd 5280 | . . . 4 ⊢ (𝜑 → 𝒫 𝐴 ∈ V) |
13 | 2 | pwexd 5280 | . . . 4 ⊢ (𝜑 → 𝒫 𝐵 ∈ V) |
14 | simprl 769 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑥 ⊆ 𝐴) | |
15 | velpw 4544 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
16 | 14, 15 | sylibr 236 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑥 ∈ 𝒫 𝐴) |
17 | simprr 771 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑦 ⊆ 𝐵) | |
18 | velpw 4544 | . . . . 5 ⊢ (𝑦 ∈ 𝒫 𝐵 ↔ 𝑦 ⊆ 𝐵) | |
19 | 17, 18 | sylibr 236 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) → 𝑦 ∈ 𝒫 𝐵) |
20 | 12, 13, 16, 19 | opabex2 7755 | . . 3 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)} ∈ V) |
21 | simpl 485 | . . . . 5 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦) → (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)) | |
22 | 21 | ssopab2i 5437 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)} ⊆ {〈𝑥, 𝑦〉 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)} |
23 | 22 | a1i 11 | . . 3 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)} ⊆ {〈𝑥, 𝑦〉 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)}) |
24 | 20, 23 | ssexd 5228 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)} ∈ V) |
25 | 3, 8, 11, 24 | fvmptd 6775 | 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 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 𝒫 cpw 4539 {copab 5128 × cxp 5553 “ cima 5558 ‘cfv 6355 (class class class)co 7156 𝒫*cimdir 34473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-imdir 34474 |
This theorem is referenced by: bj-imdirval3 34477 bj-imdirid 34478 |
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