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Mirrors > Home > MPE Home > Th. List > f1opw | Structured version Visualization version GIF version |
Description: A one-to-one mapping induces a one-to-one mapping on power sets. (Contributed by Stefan O'Rear, 18-Nov-2014.) (Revised by Mario Carneiro, 26-Jun-2015.) |
Ref | Expression |
---|---|
f1opw | ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏)):𝒫 𝐴–1-1-onto→𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1-onto→𝐵) | |
2 | dff1o3 6848 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–onto→𝐵 ∧ Fun ◡𝐹)) | |
3 | vex 3475 | . . . 4 ⊢ 𝑎 ∈ V | |
4 | 3 | funimaex 6644 | . . 3 ⊢ (Fun ◡𝐹 → (◡𝐹 “ 𝑎) ∈ V) |
5 | 2, 4 | simplbiim 503 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 “ 𝑎) ∈ V) |
6 | f1ofun 6844 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹) | |
7 | vex 3475 | . . . 4 ⊢ 𝑏 ∈ V | |
8 | 7 | funimaex 6644 | . . 3 ⊢ (Fun 𝐹 → (𝐹 “ 𝑏) ∈ V) |
9 | 6, 8 | syl 17 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝐹 “ 𝑏) ∈ V) |
10 | 1, 5, 9 | f1opw2 7680 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏)):𝒫 𝐴–1-1-onto→𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3471 𝒫 cpw 4604 ↦ cmpt 5233 ◡ccnv 5679 “ cima 5683 Fun wfun 6545 –onto→wfo 6549 –1-1-onto→wf1o 6550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pr 5431 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 |
This theorem is referenced by: ackbij2lem2 10269 |
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