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Theorem fsovcnvd 38834
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 38832 . 2 (𝜑𝐺:(𝒫 𝐵𝑚 𝐴)⟶(𝒫 𝐴𝑚 𝐵))
6 fsovcnvlem.h . . 3 𝐻 = (𝐵𝑂𝐴)
71, 3, 2, 6fsovfd 38832 . 2 (𝜑𝐻:(𝒫 𝐴𝑚 𝐵)⟶(𝒫 𝐵𝑚 𝐴))
81, 2, 3, 4, 6fsovcnvlem 38833 . 2 (𝜑 → (𝐻𝐺) = ( I ↾ (𝒫 𝐵𝑚 𝐴)))
91, 3, 2, 6, 4fsovcnvlem 38833 . 2 (𝜑 → (𝐺𝐻) = ( I ↾ (𝒫 𝐴𝑚 𝐵)))
105, 7, 8, 92fcoidinvd 6693 1 (𝜑𝐺 = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  {crab 3065  Vcvv 3351  𝒫 cpw 4297  cmpt 4863  ccnv 5248  cfv 6031  (class class class)co 6793  cmpt2 6795  𝑚 cmap 8009
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-8 2147  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-pow 4974  ax-pr 5034  ax-un 7096
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-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  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-uni 4575  df-iun 4656  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-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-map 8011
This theorem is referenced by:  fsovcnvfvd  38835  fsovf1od  38836  ntrneicnv  38902  clsneicnv  38929  neicvgnvo  38939  neicvgel1  38943
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