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Theorem f1opw 7690
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.)
Assertion
Ref Expression
f1opw (𝐹:𝐴1-1-onto𝐵 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏)):𝒫 𝐴1-1-onto→𝒫 𝐵)
Distinct variable groups:   𝐴,𝑏   𝐵,𝑏   𝐹,𝑏

Proof of Theorem f1opw
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 id 22 . 2 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1-onto𝐵)
2 dff1o3 6853 . . 3 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴onto𝐵 ∧ Fun 𝐹))
3 vex 3483 . . . 4 𝑎 ∈ V
43funimaex 6654 . . 3 (Fun 𝐹 → (𝐹𝑎) ∈ V)
52, 4simplbiim 504 . 2 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝑎) ∈ V)
6 f1ofun 6849 . . 3 (𝐹:𝐴1-1-onto𝐵 → Fun 𝐹)
7 vex 3483 . . . 4 𝑏 ∈ V
87funimaex 6654 . . 3 (Fun 𝐹 → (𝐹𝑏) ∈ V)
96, 8syl 17 . 2 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝑏) ∈ V)
101, 5, 9f1opw2 7689 1 (𝐹:𝐴1-1-onto𝐵 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏)):𝒫 𝐴1-1-onto→𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3479  𝒫 cpw 4599  cmpt 5224  ccnv 5683  cima 5687  Fun wfun 6554  ontowfo 6558  1-1-ontowf1o 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567
This theorem is referenced by:  ackbij2lem2  10280
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