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Mirrors > Home > ILE Home > Th. List > f1opw | 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 19 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1-onto→𝐵) | |
2 | dff1o3 5341 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–onto→𝐵 ∧ Fun ◡𝐹)) | |
3 | 2 | simprbi 273 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun ◡𝐹) |
4 | vex 2663 | . . . 4 ⊢ 𝑎 ∈ V | |
5 | 4 | funimaex 5178 | . . 3 ⊢ (Fun ◡𝐹 → (◡𝐹 “ 𝑎) ∈ V) |
6 | 3, 5 | syl 14 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 “ 𝑎) ∈ V) |
7 | f1ofun 5337 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹) | |
8 | vex 2663 | . . . 4 ⊢ 𝑏 ∈ V | |
9 | 8 | funimaex 5178 | . . 3 ⊢ (Fun 𝐹 → (𝐹 “ 𝑏) ∈ V) |
10 | 7, 9 | syl 14 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝐹 “ 𝑏) ∈ V) |
11 | 1, 6, 10 | f1opw2 5944 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏)):𝒫 𝐴–1-1-onto→𝒫 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1465 Vcvv 2660 𝒫 cpw 3480 ↦ cmpt 3959 ◡ccnv 4508 “ cima 4512 Fun wfun 5087 –onto→wfo 5091 –1-1-onto→wf1o 5092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 |
This theorem is referenced by: (None) |
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