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Theorem fsovd 39141
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 4383 . . . . . 6 (𝑏 = 𝐵 → 𝒫 𝑏 = 𝒫 𝐵)
43adantl 475 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝒫 𝑏 = 𝒫 𝐵)
5 simpl 476 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝑎 = 𝐴)
64, 5oveq12d 6928 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝒫 𝑏𝑚 𝑎) = (𝒫 𝐵𝑚 𝐴))
7 simpr 479 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝑏 = 𝐵)
8 rabeq 3405 . . . . . 6 (𝑎 = 𝐴 → {𝑥𝑎𝑦 ∈ (𝑓𝑥)} = {𝑥𝐴𝑦 ∈ (𝑓𝑥)})
98adantr 474 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → {𝑥𝑎𝑦 ∈ (𝑓𝑥)} = {𝑥𝐴𝑦 ∈ (𝑓𝑥)})
107, 9mpteq12dv 4958 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)}) = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}))
116, 10mpteq12dv 4958 . . 3 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑓 ∈ (𝒫 𝑏𝑚 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})) = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})))
1211adantl 475 . 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 3431 . 2 (𝜑𝐴 ∈ V)
15 fsovd.b . . 3 (𝜑𝐵𝑊)
1615elexd 3431 . 2 (𝜑𝐵 ∈ V)
17 ovex 6942 . . . 4 (𝒫 𝐵𝑚 𝐴) ∈ V
1817mptex 6747 . . 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 7053 1 (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  {crab 3121  Vcvv 3414  𝒫 cpw 4380  cmpt 4954  cfv 6127  (class class class)co 6910  cmpt2 6912  𝑚 cmap 8127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915
This theorem is referenced by:  fsovrfovd  39142  fsovfvd  39143  fsovfd  39145  fsovcnvlem  39146
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