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Theorem mptmpoopabovd 8107
Description: The operation value of a function value of a collection of ordered pairs of related elements. (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.) Add disjoint variable condition on 𝐷, 𝑓, to remove hypotheses. (Revised by SN, 13-Dec-2024.)
Hypotheses
Ref Expression
mptmpoopabbrd.g (𝜑𝐺𝑊)
mptmpoopabbrd.x (𝜑𝑋 ∈ (𝐴𝐺))
mptmpoopabbrd.y (𝜑𝑌 ∈ (𝐵𝐺))
mptmpoopabovd.m 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝑓(𝑎(𝐶𝑔)𝑏)𝑓(𝐷𝑔))}))
Assertion
Ref Expression
mptmpoopabovd (𝜑 → (𝑋(𝑀𝐺)𝑌) = {⟨𝑓, ⟩ ∣ (𝑓(𝑋(𝐶𝐺)𝑌)𝑓(𝐷𝐺))})
Distinct variable groups:   𝐴,𝑎,𝑏,𝑔   𝐵,𝑎,𝑏,𝑔   𝐷,𝑎,𝑏,𝑓,𝑔,   𝐺,𝑎,𝑏,𝑓,𝑔,   𝑔,𝑊   𝑋,𝑎,𝑏,𝑓,𝑔,   𝑌,𝑎,𝑏,𝑓,𝑔,   𝜑,𝑓,   𝐶,𝑎,𝑏,𝑔
Allowed substitution hints:   𝜑(𝑔,𝑎,𝑏)   𝐴(𝑓,)   𝐵(𝑓,)   𝐶(𝑓,)   𝑀(𝑓,𝑔,,𝑎,𝑏)   𝑊(𝑓,,𝑎,𝑏)

Proof of Theorem mptmpoopabovd
StepHypRef Expression
1 mptmpoopabbrd.g . 2 (𝜑𝐺𝑊)
2 mptmpoopabbrd.x . 2 (𝜑𝑋 ∈ (𝐴𝐺))
3 mptmpoopabbrd.y . 2 (𝜑𝑌 ∈ (𝐵𝐺))
4 oveq12 7440 . . 3 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝑎(𝐶𝐺)𝑏) = (𝑋(𝐶𝐺)𝑌))
54breqd 5154 . 2 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝑓(𝑎(𝐶𝐺)𝑏)𝑓(𝑋(𝐶𝐺)𝑌)))
6 fveq2 6906 . . . 4 (𝑔 = 𝐺 → (𝐶𝑔) = (𝐶𝐺))
76oveqd 7448 . . 3 (𝑔 = 𝐺 → (𝑎(𝐶𝑔)𝑏) = (𝑎(𝐶𝐺)𝑏))
87breqd 5154 . 2 (𝑔 = 𝐺 → (𝑓(𝑎(𝐶𝑔)𝑏)𝑓(𝑎(𝐶𝐺)𝑏)))
9 mptmpoopabovd.m . 2 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝑓(𝑎(𝐶𝑔)𝑏)𝑓(𝐷𝑔))}))
101, 2, 3, 5, 8, 9mptmpoopabbrd 8105 1 (𝜑 → (𝑋(𝑀𝐺)𝑌) = {⟨𝑓, ⟩ ∣ (𝑓(𝑋(𝐶𝐺)𝑌)𝑓(𝐷𝐺))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480   class class class wbr 5143  {copab 5205  cmpt 5225  cfv 6561  (class class class)co 7431  cmpo 7433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015
This theorem is referenced by:  wksonproplem  29722  trlsonfval  29724  pthsonfval  29760  spthson  29761
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