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Mirrors > Home > ILE Home > Th. List > pw2f1odc | 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) |
pw2f1odc.4 | ⊢ (𝜑 → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝒫 𝐴DECID 𝑝 ∈ 𝑞) |
pw2f1o.5 | ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵))) |
Ref | Expression |
---|---|
pw2f1odc | ⊢ (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 2189 | . . . 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 | pw2f1odc.4 | . . . . . 6 ⊢ (𝜑 → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝒫 𝐴DECID 𝑝 ∈ 𝑞) | |
8 | 3, 4, 5, 6, 7 | pw2f1odclem 6877 | . . . . 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})))) |
9 | 8 | biimpa 296 | . . . 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}))) |
10 | 2, 9 | mpanr2 438 | . . 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}))) |
11 | 10 | simpld 112 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) → (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵)) ∈ ({𝐵, 𝐶} ↑𝑚 𝐴)) |
12 | vex 2759 | . . . . 5 ⊢ 𝑦 ∈ V | |
13 | 12 | cnvex 5192 | . . . 4 ⊢ ◡𝑦 ∈ V |
14 | 13 | imaex 5008 | . . 3 ⊢ (◡𝑦 “ {𝐶}) ∈ V |
15 | 14 | a1i 9 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ({𝐵, 𝐶} ↑𝑚 𝐴)) → (◡𝑦 “ {𝐶}) ∈ V) |
16 | 3, 4, 5, 6, 7 | pw2f1odclem 6877 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵))) ↔ (𝑦 ∈ ({𝐵, 𝐶} ↑𝑚 𝐴) ∧ 𝑥 = (◡𝑦 “ {𝐶})))) |
17 | 1, 11, 15, 16 | f1od 6109 | 1 ⊢ (𝜑 → 𝐹:𝒫 𝐴–1-1-onto→({𝐵, 𝐶} ↑𝑚 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 DECID wdc 835 = wceq 1364 ∈ wcel 2160 ≠ wne 2360 ∀wral 2468 Vcvv 2756 ifcif 3553 𝒫 cpw 3597 {csn 3614 {cpr 3615 ↦ cmpt 4086 ◡ccnv 4650 “ cima 4654 –1-1-onto→wf1o 5241 (class class class)co 5906 ↑𝑚 cmap 6689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4143 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 |
This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2758 df-sbc 2982 df-csb 3077 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-if 3554 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-br 4026 df-opab 4087 df-mpt 4088 df-id 4318 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-rn 4662 df-res 4663 df-ima 4664 df-iota 5203 df-fun 5244 df-fn 5245 df-f 5246 df-f1 5247 df-fo 5248 df-f1o 5249 df-fv 5250 df-ov 5909 df-oprab 5910 df-mpo 5911 df-map 6691 |
This theorem is referenced by: exmidpw2en 6955 |
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