Theorem mptmpoopabovdOLD 8063

Description: Obsolete version of mptmpoopabovd 8061 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	⊢ (𝜑 → 𝐺 ∈ 𝑊)
mptmpoopabbrdOLD.x	⊢ (𝜑 → 𝑋 ∈ (𝐴‘𝐺))
mptmpoopabbrdOLD.y	⊢ (𝜑 → 𝑌 ∈ (𝐵‘𝐺))
mptmpoopabbrdOLD.v	⊢ (𝜑 → {⟨𝑓, ℎ⟩ ∣ 𝜓} ∈ 𝑉)
mptmpoopabbrdOLD.r	⊢ ((𝜑 ∧ 𝑓(𝐷‘𝐺)ℎ) → 𝜓)
mptmpoopabovdOLD.m	⊢ 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {⟨𝑓, ℎ⟩ ∣ (𝑓(𝑎(𝐶‘𝑔)𝑏)ℎ ∧ 𝑓(𝐷‘𝑔)ℎ)}))

Assertion

Ref	Expression
mptmpoopabovdOLD	⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝑓(𝑋(𝐶‘𝐺)𝑌)ℎ ∧ 𝑓(𝐷‘𝐺)ℎ)})

Distinct variable groups:   𝐴,𝑎,𝑏,𝑔   𝐵,𝑎,𝑏,𝑔   𝐷,𝑎,𝑏,𝑔   𝐺,𝑎,𝑏,𝑓,𝑔,ℎ   𝑔,𝑊   𝑋,𝑎,𝑏,𝑓,𝑔,ℎ   𝑌,𝑎,𝑏,𝑓,𝑔,ℎ   𝜑,𝑓,ℎ   𝐶,𝑎,𝑏,𝑔

Allowed substitution hints:   𝜑(𝑔,𝑎,𝑏)   𝜓(𝑓,𝑔,ℎ,𝑎,𝑏)   𝐴(𝑓,ℎ)   𝐵(𝑓,ℎ)   𝐶(𝑓,ℎ)   𝐷(𝑓,ℎ)   𝑀(𝑓,𝑔,ℎ,𝑎,𝑏)   𝑉(𝑓,𝑔,ℎ,𝑎,𝑏)   𝑊(𝑓,ℎ,𝑎,𝑏)

Proof of Theorem mptmpoopabovdOLD

Step	Hyp	Ref	Expression
1		mptmpoopabbrdOLD.g	. 2 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
2		mptmpoopabbrdOLD.x	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝐴‘𝐺))
3		mptmpoopabbrdOLD.y	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝐵‘𝐺))
4		mptmpoopabbrdOLD.v	. 2 ⊢ (𝜑 → {⟨𝑓, ℎ⟩ ∣ 𝜓} ∈ 𝑉)
5		mptmpoopabbrdOLD.r	. 2 ⊢ ((𝜑 ∧ 𝑓(𝐷‘𝐺)ℎ) → 𝜓)
6		oveq12 7396	. . 3 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝑎(𝐶‘𝐺)𝑏) = (𝑋(𝐶‘𝐺)𝑌))
7	6	breqd 5118	. 2 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝑓(𝑎(𝐶‘𝐺)𝑏)ℎ ↔ 𝑓(𝑋(𝐶‘𝐺)𝑌)ℎ))
8		fveq2 6858	. . . 4 ⊢ (𝑔 = 𝐺 → (𝐶‘𝑔) = (𝐶‘𝐺))
9	8	oveqd 7404	. . 3 ⊢ (𝑔 = 𝐺 → (𝑎(𝐶‘𝑔)𝑏) = (𝑎(𝐶‘𝐺)𝑏))
10	9	breqd 5118	. 2 ⊢ (𝑔 = 𝐺 → (𝑓(𝑎(𝐶‘𝑔)𝑏)ℎ ↔ 𝑓(𝑎(𝐶‘𝐺)𝑏)ℎ))
11		mptmpoopabovdOLD.m	. 2 ⊢ 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {⟨𝑓, ℎ⟩ ∣ (𝑓(𝑎(𝐶‘𝑔)𝑏)ℎ ∧ 𝑓(𝐷‘𝑔)ℎ)}))
12	1, 2, 3, 4, 5, 7, 10, 11	mptmpoopabbrdOLDOLD 8062	1 ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝑓(𝑋(𝐶‘𝐺)𝑌)ℎ ∧ 𝑓(𝐷‘𝐺)ℎ)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447   class class class wbr 5107  {copab 5169   ↦ cmpt 5188  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969

This theorem is referenced by:  wksonproplemOLD  29633