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Theorem fsovfd 43584
Description: The operator, (𝐴𝑂𝐵), 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, 𝐴 and 𝐵, gives a function between two sets of functions. (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
fsovfd (𝜑𝐺:(𝒫 𝐵m 𝐴)⟶(𝒫 𝐴m 𝐵))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑓   𝑥,𝐴,𝑎,𝑏   𝑦,𝐴,𝑎,𝑏   𝐵,𝑎,𝑏,𝑓   𝑦,𝐵   𝜑,𝑎,𝑏,𝑓   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐵(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 fsovfd
StepHypRef Expression
1 fsovfvd.g . . 3 𝐺 = (𝐴𝑂𝐵)
2 fsovd.fs . . . 4 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏m 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))
3 fsovd.a . . . 4 (𝜑𝐴𝑉)
4 fsovd.b . . . 4 (𝜑𝐵𝑊)
52, 3, 4fsovd 43580 . . 3 (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})))
61, 5eqtrid 2777 . 2 (𝜑𝐺 = (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})))
7 ssrab2 4073 . . . . . . . 8 {𝑥𝐴𝑦 ∈ (𝑓𝑥)} ⊆ 𝐴
87a1i 11 . . . . . . 7 (𝜑 → {𝑥𝐴𝑦 ∈ (𝑓𝑥)} ⊆ 𝐴)
93, 8sselpwd 5329 . . . . . 6 (𝜑 → {𝑥𝐴𝑦 ∈ (𝑓𝑥)} ∈ 𝒫 𝐴)
109adantr 479 . . . . 5 ((𝜑𝑦𝐵) → {𝑥𝐴𝑦 ∈ (𝑓𝑥)} ∈ 𝒫 𝐴)
1110fmpttd 7124 . . . 4 (𝜑 → (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}):𝐵⟶𝒫 𝐴)
123pwexd 5379 . . . . 5 (𝜑 → 𝒫 𝐴 ∈ V)
1312, 4elmapd 8859 . . . 4 (𝜑 → ((𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}) ∈ (𝒫 𝐴m 𝐵) ↔ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}):𝐵⟶𝒫 𝐴))
1411, 13mpbird 256 . . 3 (𝜑 → (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}) ∈ (𝒫 𝐴m 𝐵))
1514adantr 479 . 2 ((𝜑𝑓 ∈ (𝒫 𝐵m 𝐴)) → (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}) ∈ (𝒫 𝐴m 𝐵))
166, 15fmpt3d 7125 1 (𝜑𝐺:(𝒫 𝐵m 𝐴)⟶(𝒫 𝐴m 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  {crab 3418  Vcvv 3461  wss 3944  𝒫 cpw 4604  cmpt 5232  wf 6545  cfv 6549  (class class class)co 7419  cmpo 7421  m cmap 8845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8847
This theorem is referenced by:  fsovcnvd  43586  fsovf1od  43588  clsneiel1  43680  neicvgmex  43689  neicvgel1  43691
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