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Theorem mptmpoopabovd 7784
 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.)
Hypotheses
Ref Expression
mptmpoopabbrd.g (𝜑𝐺𝑊)
mptmpoopabbrd.x (𝜑𝑋 ∈ (𝐴𝐺))
mptmpoopabbrd.y (𝜑𝑌 ∈ (𝐵𝐺))
mptmpoopabbrd.v (𝜑 → {⟨𝑓, ⟩ ∣ 𝜓} ∈ 𝑉)
mptmpoopabbrd.r ((𝜑𝑓(𝐷𝐺)) → 𝜓)
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,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)   𝜓(𝑓,𝑔,,𝑎,𝑏)   𝐴(𝑓,)   𝐵(𝑓,)   𝐶(𝑓,)   𝐷(𝑓,)   𝑀(𝑓,𝑔,,𝑎,𝑏)   𝑉(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 mptmpoopabbrd.v . 2 (𝜑 → {⟨𝑓, ⟩ ∣ 𝜓} ∈ 𝑉)
5 mptmpoopabbrd.r . 2 ((𝜑𝑓(𝐷𝐺)) → 𝜓)
6 oveq12 7159 . . 3 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝑎(𝐶𝐺)𝑏) = (𝑋(𝐶𝐺)𝑌))
76breqd 5043 . 2 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝑓(𝑎(𝐶𝐺)𝑏)𝑓(𝑋(𝐶𝐺)𝑌)))
8 fveq2 6658 . . . 4 (𝑔 = 𝐺 → (𝐶𝑔) = (𝐶𝐺))
98oveqd 7167 . . 3 (𝑔 = 𝐺 → (𝑎(𝐶𝑔)𝑏) = (𝑎(𝐶𝐺)𝑏))
109breqd 5043 . 2 (𝑔 = 𝐺 → (𝑓(𝑎(𝐶𝑔)𝑏)𝑓(𝑎(𝐶𝐺)𝑏)))
11 mptmpoopabovd.m . 2 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝑓(𝑎(𝐶𝑔)𝑏)𝑓(𝐷𝑔))}))
121, 2, 3, 4, 5, 7, 10, 11mptmpoopabbrd 7783 1 (𝜑 → (𝑋(𝑀𝐺)𝑌) = {⟨𝑓, ⟩ ∣ (𝑓(𝑋(𝐶𝐺)𝑌)𝑓(𝐷𝐺))})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409   class class class wbr 5032  {copab 5094   ↦ cmpt 5112  ‘cfv 6335  (class class class)co 7150   ∈ cmpo 7152 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7693  df-2nd 7694 This theorem is referenced by:  wksonproplem  27593  trlsonfval  27594  pthsonfval  27628  spthson  27629
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