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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 6854 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–onto→𝐵 ∧ Fun ◡𝐹)) | |
| 3 | vex 3484 | . . . 4 ⊢ 𝑎 ∈ V | |
| 4 | 3 | funimaex 6655 | . . 3 ⊢ (Fun ◡𝐹 → (◡𝐹 “ 𝑎) ∈ V) | 
| 5 | 2, 4 | simplbiim 504 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 “ 𝑎) ∈ V) | 
| 6 | f1ofun 6850 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹) | |
| 7 | vex 3484 | . . . 4 ⊢ 𝑏 ∈ V | |
| 8 | 7 | funimaex 6655 | . . 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 7688 | 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 2108 Vcvv 3480 𝒫 cpw 4600 ↦ cmpt 5225 ◡ccnv 5684 “ cima 5688 Fun wfun 6555 –onto→wfo 6559 –1-1-onto→wf1o 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 | 
| This theorem is referenced by: ackbij2lem2 10279 | 
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