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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 6596 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–onto→𝐵 ∧ Fun ◡𝐹)) | |
3 | vex 3444 | . . . 4 ⊢ 𝑎 ∈ V | |
4 | 3 | funimaex 6411 | . . 3 ⊢ (Fun ◡𝐹 → (◡𝐹 “ 𝑎) ∈ V) |
5 | 2, 4 | simplbiim 508 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 “ 𝑎) ∈ V) |
6 | f1ofun 6592 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹) | |
7 | vex 3444 | . . . 4 ⊢ 𝑏 ∈ V | |
8 | 7 | funimaex 6411 | . . 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 7380 | 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 2111 Vcvv 3441 𝒫 cpw 4497 ↦ cmpt 5110 ◡ccnv 5518 “ cima 5522 Fun wfun 6318 –onto→wfo 6322 –1-1-onto→wf1o 6323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 |
This theorem is referenced by: ackbij2lem2 9651 |
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