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Theorem fsovf1od 37330
 Description: The value of (𝐴𝑂𝐵) is a bijection, where 𝑂 is the operator which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets. (Contributed by RP, 27-Apr-2021.)
Hypotheses
Ref Expression
fsovd.fs 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))
fsovd.a (𝜑𝐴𝑉)
fsovd.b (𝜑𝐵𝑊)
fsovfvd.g 𝐺 = (𝐴𝑂𝐵)
Assertion
Ref Expression
fsovf1od (𝜑𝐺:(𝒫 𝐵𝑚 𝐴)–1-1-onto→(𝒫 𝐴𝑚 𝐵))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑓,𝑥,𝑦   𝐵,𝑎,𝑏,𝑓,𝑥,𝑦   𝜑,𝑎,𝑏,𝑓,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐺(𝑥,𝑦,𝑓,𝑎,𝑏)   𝑂(𝑥,𝑦,𝑓,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑓,𝑎,𝑏)   𝑊(𝑥,𝑦,𝑓,𝑎,𝑏)

Proof of Theorem fsovf1od
StepHypRef Expression
1 fsovd.fs . . . 4 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))
2 fsovd.a . . . 4 (𝜑𝐴𝑉)
3 fsovd.b . . . 4 (𝜑𝐵𝑊)
4 fsovfvd.g . . . 4 𝐺 = (𝐴𝑂𝐵)
51, 2, 3, 4fsovfd 37326 . . 3 (𝜑𝐺:(𝒫 𝐵𝑚 𝐴)⟶(𝒫 𝐴𝑚 𝐵))
65ffnd 5959 . 2 (𝜑𝐺 Fn (𝒫 𝐵𝑚 𝐴))
7 eqid 2610 . . . . 5 (𝐵𝑂𝐴) = (𝐵𝑂𝐴)
81, 3, 2, 7fsovfd 37326 . . . 4 (𝜑 → (𝐵𝑂𝐴):(𝒫 𝐴𝑚 𝐵)⟶(𝒫 𝐵𝑚 𝐴))
98ffnd 5959 . . 3 (𝜑 → (𝐵𝑂𝐴) Fn (𝒫 𝐴𝑚 𝐵))
101, 2, 3, 4, 7fsovcnvd 37328 . . . 4 (𝜑𝐺 = (𝐵𝑂𝐴))
1110fneq1d 5895 . . 3 (𝜑 → (𝐺 Fn (𝒫 𝐴𝑚 𝐵) ↔ (𝐵𝑂𝐴) Fn (𝒫 𝐴𝑚 𝐵)))
129, 11mpbird 246 . 2 (𝜑𝐺 Fn (𝒫 𝐴𝑚 𝐵))
13 dff1o4 6058 . 2 (𝐺:(𝒫 𝐵𝑚 𝐴)–1-1-onto→(𝒫 𝐴𝑚 𝐵) ↔ (𝐺 Fn (𝒫 𝐵𝑚 𝐴) ∧ 𝐺 Fn (𝒫 𝐴𝑚 𝐵)))
146, 12, 13sylanbrc 695 1 (𝜑𝐺:(𝒫 𝐵𝑚 𝐴)–1-1-onto→(𝒫 𝐴𝑚 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  𝒫 cpw 4108   ↦ cmpt 4643  ◡ccnv 5037   Fn wfn 5799  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   ↑𝑚 cmap 7744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746 This theorem is referenced by:  ntrneif1o  37393  clsneif1o  37422  clsneikex  37424  clsneinex  37425  neicvgf1o  37432  neicvgmex  37435  neicvgel1  37437
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