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Theorem fsovd 40626
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 4538 . . . . . 6 (𝑏 = 𝐵 → 𝒫 𝑏 = 𝒫 𝐵)
43adantl 485 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝒫 𝑏 = 𝒫 𝐵)
5 simpl 486 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝑎 = 𝐴)
64, 5oveq12d 7167 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝒫 𝑏m 𝑎) = (𝒫 𝐵m 𝐴))
7 simpr 488 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝑏 = 𝐵)
8 rabeq 3469 . . . . . 6 (𝑎 = 𝐴 → {𝑥𝑎𝑦 ∈ (𝑓𝑥)} = {𝑥𝐴𝑦 ∈ (𝑓𝑥)})
98adantr 484 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → {𝑥𝑎𝑦 ∈ (𝑓𝑥)} = {𝑥𝐴𝑦 ∈ (𝑓𝑥)})
107, 9mpteq12dv 5137 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)}) = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}))
116, 10mpteq12dv 5137 . . 3 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑓 ∈ (𝒫 𝑏m 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})) = (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})))
1211adantl 485 . 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 3500 . 2 (𝜑𝐴 ∈ V)
15 fsovd.b . . 3 (𝜑𝐵𝑊)
1615elexd 3500 . 2 (𝜑𝐵 ∈ V)
17 ovex 7182 . . . 4 (𝒫 𝐵m 𝐴) ∈ V
1817mptex 6977 . . 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, 19ovmpod 7295 1 (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  {crab 3137  Vcvv 3480  𝒫 cpw 4522  cmpt 5132  cfv 6343  (class class class)co 7149  cmpo 7151  m cmap 8402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154
This theorem is referenced by:  fsovrfovd  40627  fsovfvd  40628  fsovfd  40630  fsovcnvlem  40631
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