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Theorem f1opw 7036
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 6284 . . . 4 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴onto𝐵 ∧ Fun 𝐹))
32simprbi 484 . . 3 (𝐹:𝐴1-1-onto𝐵 → Fun 𝐹)
4 vex 3354 . . . 4 𝑎 ∈ V
54funimaex 6116 . . 3 (Fun 𝐹 → (𝐹𝑎) ∈ V)
63, 5syl 17 . 2 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝑎) ∈ V)
7 f1ofun 6280 . . 3 (𝐹:𝐴1-1-onto𝐵 → Fun 𝐹)
8 vex 3354 . . . 4 𝑏 ∈ V
98funimaex 6116 . . 3 (Fun 𝐹 → (𝐹𝑏) ∈ V)
107, 9syl 17 . 2 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝑏) ∈ V)
111, 6, 10f1opw2 7035 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 2145  Vcvv 3351  𝒫 cpw 4297  cmpt 4863  ccnv 5248  cima 5252  Fun wfun 6025  ontowfo 6029  1-1-ontowf1o 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038
This theorem is referenced by:  ackbij2lem2  9264
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