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Theorem fsovcnvd 40633
Description: The value of the converse (𝐴𝑂𝐵) is (𝐵𝑂𝐴), 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, gives a family of functions that include their own inverse. (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)
fsovcnvlem.h 𝐻 = (𝐵𝑂𝐴)
Assertion
Ref Expression
fsovcnvd (𝜑𝐺 = 𝐻)
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)   𝐻(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 fsovcnvd
StepHypRef Expression
1 fsovd.fs . . 3 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏m 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))
2 fsovd.a . . 3 (𝜑𝐴𝑉)
3 fsovd.b . . 3 (𝜑𝐵𝑊)
4 fsovfvd.g . . 3 𝐺 = (𝐴𝑂𝐵)
51, 2, 3, 4fsovfd 40631 . 2 (𝜑𝐺:(𝒫 𝐵m 𝐴)⟶(𝒫 𝐴m 𝐵))
6 fsovcnvlem.h . . 3 𝐻 = (𝐵𝑂𝐴)
71, 3, 2, 6fsovfd 40631 . 2 (𝜑𝐻:(𝒫 𝐴m 𝐵)⟶(𝒫 𝐵m 𝐴))
81, 2, 3, 4, 6fsovcnvlem 40632 . 2 (𝜑 → (𝐻𝐺) = ( I ↾ (𝒫 𝐵m 𝐴)))
91, 3, 2, 6, 4fsovcnvlem 40632 . 2 (𝜑 → (𝐺𝐻) = ( I ↾ (𝒫 𝐴m 𝐵)))
105, 7, 8, 92fcoidinvd 7044 1 (𝜑𝐺 = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  {crab 3137  Vcvv 3481  𝒫 cpw 4523  cmpt 5133  ccnv 5542  cfv 6344  (class class class)co 7150  cmpo 7152  m cmap 8403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7685  df-2nd 7686  df-map 8405
This theorem is referenced by:  fsovcnvfvd  40634  fsovf1od  40635  ntrneicnv  40701  clsneicnv  40728  neicvgnvo  40738  neicvgel1  40742
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