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Theorem fsovf1od 43978
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 43974 . . 3 (𝜑𝐺:(𝒫 𝐵m 𝐴)⟶(𝒫 𝐴m 𝐵))
65ffnd 6748 . 2 (𝜑𝐺 Fn (𝒫 𝐵m 𝐴))
7 eqid 2740 . . . . 5 (𝐵𝑂𝐴) = (𝐵𝑂𝐴)
81, 3, 2, 7fsovfd 43974 . . . 4 (𝜑 → (𝐵𝑂𝐴):(𝒫 𝐴m 𝐵)⟶(𝒫 𝐵m 𝐴))
98ffnd 6748 . . 3 (𝜑 → (𝐵𝑂𝐴) Fn (𝒫 𝐴m 𝐵))
101, 2, 3, 4, 7fsovcnvd 43976 . . . 4 (𝜑𝐺 = (𝐵𝑂𝐴))
1110fneq1d 6672 . . 3 (𝜑 → (𝐺 Fn (𝒫 𝐴m 𝐵) ↔ (𝐵𝑂𝐴) Fn (𝒫 𝐴m 𝐵)))
129, 11mpbird 257 . 2 (𝜑𝐺 Fn (𝒫 𝐴m 𝐵))
13 dff1o4 6870 . 2 (𝐺:(𝒫 𝐵m 𝐴)–1-1-onto→(𝒫 𝐴m 𝐵) ↔ (𝐺 Fn (𝒫 𝐵m 𝐴) ∧ 𝐺 Fn (𝒫 𝐴m 𝐵)))
146, 12, 13sylanbrc 582 1 (𝜑𝐺:(𝒫 𝐵m 𝐴)–1-1-onto→(𝒫 𝐴m 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2108  {crab 3443  Vcvv 3488  𝒫 cpw 4622  cmpt 5249  ccnv 5699   Fn wfn 6568  1-1-ontowf1o 6572  cfv 6573  (class class class)co 7448  cmpo 7450  m cmap 8884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-map 8886
This theorem is referenced by:  ntrneif1o  44037  clsneif1o  44066  clsneikex  44068  clsneinex  44069  neicvgf1o  44076  neicvgmex  44079  neicvgel1  44081
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