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Mirrors > Home > MPE Home > Th. List > pw2f1o | Structured version Visualization version GIF version |
Description: The power set of a set is equinumerous to set exponentiation with an unordered pair base of ordinal 2. Generalized from Proposition 10.44 of [TakeutiZaring] p. 96. (Contributed by Mario Carneiro, 6-Oct-2014.) |
Ref | Expression |
---|---|
pw2f1o.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
pw2f1o.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
pw2f1o.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑊) |
pw2f1o.4 | ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
pw2f1o.5 | ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵))) |
Ref | Expression |
---|---|
pw2f1o | ⊢ (𝜑 → 𝐹:𝒫 𝐴–1-1-onto→({𝐵, 𝐶} ↑m 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pw2f1o.5 | . 2 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵))) | |
2 | eqid 2823 | . . . 4 ⊢ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) | |
3 | pw2f1o.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | pw2f1o.2 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
5 | pw2f1o.3 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
6 | pw2f1o.4 | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
7 | 3, 4, 5, 6 | pw2f1olem 8623 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝒫 𝐴 ∧ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵))) ↔ ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) ∈ ({𝐵, 𝐶} ↑m 𝐴) ∧ 𝑥 = (◡(𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) “ {𝐶})))) |
8 | 7 | biimpa 479 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)))) → ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) ∈ ({𝐵, 𝐶} ↑m 𝐴) ∧ 𝑥 = (◡(𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) “ {𝐶}))) |
9 | 2, 8 | mpanr2 702 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) → ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) ∈ ({𝐵, 𝐶} ↑m 𝐴) ∧ 𝑥 = (◡(𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) “ {𝐶}))) |
10 | 9 | simpld 497 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) → (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) ∈ ({𝐵, 𝐶} ↑m 𝐴)) |
11 | vex 3499 | . . . . 5 ⊢ 𝑦 ∈ V | |
12 | 11 | cnvex 7632 | . . . 4 ⊢ ◡𝑦 ∈ V |
13 | 12 | imaex 7623 | . . 3 ⊢ (◡𝑦 “ {𝐶}) ∈ V |
14 | 13 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ({𝐵, 𝐶} ↑m 𝐴)) → (◡𝑦 “ {𝐶}) ∈ V) |
15 | 3, 4, 5, 6 | pw2f1olem 8623 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵))) ↔ (𝑦 ∈ ({𝐵, 𝐶} ↑m 𝐴) ∧ 𝑥 = (◡𝑦 “ {𝐶})))) |
16 | 1, 10, 14, 15 | f1od 7399 | 1 ⊢ (𝜑 → 𝐹:𝒫 𝐴–1-1-onto→({𝐵, 𝐶} ↑m 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ifcif 4469 𝒫 cpw 4541 {csn 4569 {cpr 4571 ↦ cmpt 5148 ◡ccnv 5556 “ cima 5560 –1-1-onto→wf1o 6356 (class class class)co 7158 ↑m cmap 8408 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 |
This theorem is referenced by: pw2eng 8625 indf1o 31285 |
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