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Theorem mptmpoopabovdOLD 8015
Description: Obsolete version of mptmpoopabovd 8013 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 7365 . . 3 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝑎(𝐶𝐺)𝑏) = (𝑋(𝐶𝐺)𝑌))
76breqd 5116 . 2 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝑓(𝑎(𝐶𝐺)𝑏)𝑓(𝑋(𝐶𝐺)𝑌)))
8 fveq2 6842 . . . 4 (𝑔 = 𝐺 → (𝐶𝑔) = (𝐶𝐺))
98oveqd 7373 . . 3 (𝑔 = 𝐺 → (𝑎(𝐶𝑔)𝑏) = (𝑎(𝐶𝐺)𝑏))
109breqd 5116 . 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, 11mptmpoopabbrdOLD 8014 1 (𝜑 → (𝑋(𝑀𝐺)𝑌) = {⟨𝑓, ⟩ ∣ (𝑓(𝑋(𝐶𝐺)𝑌)𝑓(𝐷𝐺))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3445   class class class wbr 5105  {copab 5167  cmpt 5188  cfv 6496  (class class class)co 7356  cmpo 7358
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7920  df-2nd 7921
This theorem is referenced by:  wksonproplemOLD  28600
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