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Theorem fsovd 37322
Description: Value of 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. (Contributed by RP, 25-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)
Assertion
Ref Expression
fsovd (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑓   𝑥,𝐴,𝑎,𝑏   𝑦,𝐴,𝑎,𝑏   𝐵,𝑎,𝑏,𝑓   𝑦,𝐵   𝜑,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓)   𝐵(𝑥)   𝑂(𝑥,𝑦,𝑓,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑓,𝑎,𝑏)   𝑊(𝑥,𝑦,𝑓,𝑎,𝑏)

Proof of Theorem fsovd
StepHypRef Expression
1 fsovd.fs . . 3 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))
21a1i 11 . 2 (𝜑𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)}))))
3 pweq 4111 . . . . . 6 (𝑏 = 𝐵 → 𝒫 𝑏 = 𝒫 𝐵)
43adantl 481 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝒫 𝑏 = 𝒫 𝐵)
5 simpl 472 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝑎 = 𝐴)
64, 5oveq12d 6567 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝒫 𝑏𝑚 𝑎) = (𝒫 𝐵𝑚 𝐴))
7 simpr 476 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝑏 = 𝐵)
8 rabeq 3166 . . . . . 6 (𝑎 = 𝐴 → {𝑥𝑎𝑦 ∈ (𝑓𝑥)} = {𝑥𝐴𝑦 ∈ (𝑓𝑥)})
98adantr 480 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → {𝑥𝑎𝑦 ∈ (𝑓𝑥)} = {𝑥𝐴𝑦 ∈ (𝑓𝑥)})
107, 9mpteq12dv 4663 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)}) = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}))
116, 10mpteq12dv 4663 . . 3 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑓 ∈ (𝒫 𝑏𝑚 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})) = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})))
1211adantl 481 . 2 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑓 ∈ (𝒫 𝑏𝑚 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})) = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})))
13 fsovd.a . . 3 (𝜑𝐴𝑉)
1413elexd 3187 . 2 (𝜑𝐴 ∈ V)
15 fsovd.b . . 3 (𝜑𝐵𝑊)
1615elexd 3187 . 2 (𝜑𝐵 ∈ V)
17 ovex 6577 . . . 4 (𝒫 𝐵𝑚 𝐴) ∈ V
1817mptex 6390 . . 3 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})) ∈ V
1918a1i 11 . 2 (𝜑 → (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})) ∈ V)
202, 12, 14, 16, 19ovmpt2d 6686 1 (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  {crab 2900  Vcvv 3173  𝒫 cpw 4108  cmpt 4643  cfv 5804  (class class class)co 6549  cmpt2 6551  𝑚 cmap 7744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554
This theorem is referenced by:  fsovrfovd  37323  fsovfvd  37324  fsovfd  37326  fsovcnvlem  37327
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