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Theorem mptmpoopabovdOLD 8108
Description: Obsolete version of mptmpoopabovd 8106 as of 13-Dec-2024. (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
mptmpoopabbrdOLD.g (𝜑𝐺𝑊)
mptmpoopabbrdOLD.x (𝜑𝑋 ∈ (𝐴𝐺))
mptmpoopabbrdOLD.y (𝜑𝑌 ∈ (𝐵𝐺))
mptmpoopabbrdOLD.v (𝜑 → {⟨𝑓, ⟩ ∣ 𝜓} ∈ 𝑉)
mptmpoopabbrdOLD.r ((𝜑𝑓(𝐷𝐺)) → 𝜓)
mptmpoopabovdOLD.m 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝑓(𝑎(𝐶𝑔)𝑏)𝑓(𝐷𝑔))}))
Assertion
Ref Expression
mptmpoopabovdOLD (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 mptmpoopabovdOLD
StepHypRef Expression
1 mptmpoopabbrdOLD.g . 2 (𝜑𝐺𝑊)
2 mptmpoopabbrdOLD.x . 2 (𝜑𝑋 ∈ (𝐴𝐺))
3 mptmpoopabbrdOLD.y . 2 (𝜑𝑌 ∈ (𝐵𝐺))
4 mptmpoopabbrdOLD.v . 2 (𝜑 → {⟨𝑓, ⟩ ∣ 𝜓} ∈ 𝑉)
5 mptmpoopabbrdOLD.r . 2 ((𝜑𝑓(𝐷𝐺)) → 𝜓)
6 oveq12 7440 . . 3 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝑎(𝐶𝐺)𝑏) = (𝑋(𝐶𝐺)𝑌))
76breqd 5159 . 2 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝑓(𝑎(𝐶𝐺)𝑏)𝑓(𝑋(𝐶𝐺)𝑌)))
8 fveq2 6907 . . . 4 (𝑔 = 𝐺 → (𝐶𝑔) = (𝐶𝐺))
98oveqd 7448 . . 3 (𝑔 = 𝐺 → (𝑎(𝐶𝑔)𝑏) = (𝑎(𝐶𝐺)𝑏))
109breqd 5159 . 2 (𝑔 = 𝐺 → (𝑓(𝑎(𝐶𝑔)𝑏)𝑓(𝑎(𝐶𝐺)𝑏)))
11 mptmpoopabovdOLD.m . 2 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝑓(𝑎(𝐶𝑔)𝑏)𝑓(𝐷𝑔))}))
121, 2, 3, 4, 5, 7, 10, 11mptmpoopabbrdOLDOLD 8107 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 1537  wcel 2106  Vcvv 3478   class class class wbr 5148  {copab 5210  cmpt 5231  cfv 6563  (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 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014
This theorem is referenced by:  wksonproplemOLD  29738
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