ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f1opw GIF version

Theorem f1opw 5945
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 19 . 2 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1-onto𝐵)
2 dff1o3 5341 . . . 4 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴onto𝐵 ∧ Fun 𝐹))
32simprbi 273 . . 3 (𝐹:𝐴1-1-onto𝐵 → Fun 𝐹)
4 vex 2663 . . . 4 𝑎 ∈ V
54funimaex 5178 . . 3 (Fun 𝐹 → (𝐹𝑎) ∈ V)
63, 5syl 14 . 2 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝑎) ∈ V)
7 f1ofun 5337 . . 3 (𝐹:𝐴1-1-onto𝐵 → Fun 𝐹)
8 vex 2663 . . . 4 𝑏 ∈ V
98funimaex 5178 . . 3 (Fun 𝐹 → (𝐹𝑏) ∈ V)
107, 9syl 14 . 2 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝑏) ∈ V)
111, 6, 10f1opw2 5944 1 (𝐹:𝐴1-1-onto𝐵 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏)):𝒫 𝐴1-1-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  ontowfo 5091  1-1-ontowf1o 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)
  Copyright terms: Public domain W3C validator