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Theorem f1opw 7613
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 6794 . . 3 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴onto𝐵 ∧ Fun 𝐹))
3 vex 3451 . . . 4 𝑎 ∈ V
43funimaex 6593 . . 3 (Fun 𝐹 → (𝐹𝑎) ∈ V)
52, 4simplbiim 506 . 2 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝑎) ∈ V)
6 f1ofun 6790 . . 3 (𝐹:𝐴1-1-onto𝐵 → Fun 𝐹)
7 vex 3451 . . . 4 𝑏 ∈ V
87funimaex 6593 . . 3 (Fun 𝐹 → (𝐹𝑏) ∈ V)
96, 8syl 17 . 2 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝑏) ∈ V)
101, 5, 9f1opw2 7612 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 3447  𝒫 cpw 4564  cmpt 5192  ccnv 5636  cima 5640  Fun wfun 6494  ontowfo 6498  1-1-ontowf1o 6499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507
This theorem is referenced by:  ackbij2lem2  10184
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