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Theorem fsovfd 44002
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 43998 . . 3 (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})))
61, 5eqtrid 2787 . 2 (𝜑𝐺 = (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})))
7 ssrab2 4090 . . . . . . . 8 {𝑥𝐴𝑦 ∈ (𝑓𝑥)} ⊆ 𝐴
87a1i 11 . . . . . . 7 (𝜑 → {𝑥𝐴𝑦 ∈ (𝑓𝑥)} ⊆ 𝐴)
93, 8sselpwd 5334 . . . . . 6 (𝜑 → {𝑥𝐴𝑦 ∈ (𝑓𝑥)} ∈ 𝒫 𝐴)
109adantr 480 . . . . 5 ((𝜑𝑦𝐵) → {𝑥𝐴𝑦 ∈ (𝑓𝑥)} ∈ 𝒫 𝐴)
1110fmpttd 7135 . . . 4 (𝜑 → (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}):𝐵⟶𝒫 𝐴)
123pwexd 5385 . . . . 5 (𝜑 → 𝒫 𝐴 ∈ V)
1312, 4elmapd 8879 . . . 4 (𝜑 → ((𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}) ∈ (𝒫 𝐴m 𝐵) ↔ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}):𝐵⟶𝒫 𝐴))
1411, 13mpbird 257 . . 3 (𝜑 → (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}) ∈ (𝒫 𝐴m 𝐵))
1514adantr 480 . 2 ((𝜑𝑓 ∈ (𝒫 𝐵m 𝐴)) → (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}) ∈ (𝒫 𝐴m 𝐵))
166, 15fmpt3d 7136 1 (𝜑𝐺:(𝒫 𝐵m 𝐴)⟶(𝒫 𝐴m 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  wss 3963  𝒫 cpw 4605  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  m cmap 8865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867
This theorem is referenced by:  fsovcnvd  44004  fsovf1od  44006  clsneiel1  44098  neicvgmex  44107  neicvgel1  44109
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