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Theorem fsovf1od 38812
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 38808 . . 3 (𝜑𝐺:(𝒫 𝐵𝑚 𝐴)⟶(𝒫 𝐴𝑚 𝐵))
65ffnd 6207 . 2 (𝜑𝐺 Fn (𝒫 𝐵𝑚 𝐴))
7 eqid 2760 . . . . 5 (𝐵𝑂𝐴) = (𝐵𝑂𝐴)
81, 3, 2, 7fsovfd 38808 . . . 4 (𝜑 → (𝐵𝑂𝐴):(𝒫 𝐴𝑚 𝐵)⟶(𝒫 𝐵𝑚 𝐴))
98ffnd 6207 . . 3 (𝜑 → (𝐵𝑂𝐴) Fn (𝒫 𝐴𝑚 𝐵))
101, 2, 3, 4, 7fsovcnvd 38810 . . . 4 (𝜑𝐺 = (𝐵𝑂𝐴))
1110fneq1d 6142 . . 3 (𝜑 → (𝐺 Fn (𝒫 𝐴𝑚 𝐵) ↔ (𝐵𝑂𝐴) Fn (𝒫 𝐴𝑚 𝐵)))
129, 11mpbird 247 . 2 (𝜑𝐺 Fn (𝒫 𝐴𝑚 𝐵))
13 dff1o4 6306 . 2 (𝐺:(𝒫 𝐵𝑚 𝐴)–1-1-onto→(𝒫 𝐴𝑚 𝐵) ↔ (𝐺 Fn (𝒫 𝐵𝑚 𝐴) ∧ 𝐺 Fn (𝒫 𝐴𝑚 𝐵)))
146, 12, 13sylanbrc 701 1 (𝜑𝐺:(𝒫 𝐵𝑚 𝐴)–1-1-onto→(𝒫 𝐴𝑚 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  𝒫 cpw 4302  cmpt 4881  ccnv 5265   Fn wfn 6044  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6813  cmpt2 6815  𝑚 cmap 8023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025
This theorem is referenced by:  ntrneif1o  38875  clsneif1o  38904  clsneikex  38906  clsneinex  38907  neicvgf1o  38914  neicvgmex  38917  neicvgel1  38919
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