![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6399 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–onto→𝐵 ∧ Fun ◡𝐹)) | |
3 | vex 3401 | . . . 4 ⊢ 𝑎 ∈ V | |
4 | 3 | funimaex 6223 | . . 3 ⊢ (Fun ◡𝐹 → (◡𝐹 “ 𝑎) ∈ V) |
5 | 2, 4 | simplbiim 500 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 “ 𝑎) ∈ V) |
6 | f1ofun 6395 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹) | |
7 | vex 3401 | . . . 4 ⊢ 𝑏 ∈ V | |
8 | 7 | funimaex 6223 | . . 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 7167 | 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 2107 Vcvv 3398 𝒫 cpw 4379 ↦ cmpt 4967 ◡ccnv 5356 “ cima 5360 Fun wfun 6131 –onto→wfo 6135 –1-1-onto→wf1o 6136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 |
This theorem is referenced by: ackbij2lem2 9399 |
Copyright terms: Public domain | W3C validator |