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Theorem f1opw 7392
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 6610 . . 3 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴onto𝐵 ∧ Fun 𝐹))
3 vex 3483 . . . 4 𝑎 ∈ V
43funimaex 6430 . . 3 (Fun 𝐹 → (𝐹𝑎) ∈ V)
52, 4simplbiim 508 . 2 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝑎) ∈ V)
6 f1ofun 6606 . . 3 (𝐹:𝐴1-1-onto𝐵 → Fun 𝐹)
7 vex 3483 . . . 4 𝑏 ∈ V
87funimaex 6430 . . 3 (Fun 𝐹 → (𝐹𝑏) ∈ V)
96, 8syl 17 . 2 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝑏) ∈ V)
101, 5, 9f1opw2 7391 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 2115  Vcvv 3480  𝒫 cpw 4522  cmpt 5133  ccnv 5542  cima 5546  Fun wfun 6338  ontowfo 6342  1-1-ontowf1o 6343
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pr 5318
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351
This theorem is referenced by:  ackbij2lem2  9656
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