Theorem mptmpoopabovd 8088

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, ℎ to remove hypotheses. (Revised by SN, 13-Dec-2024.)

Hypotheses

Ref	Expression
mptmpoopabbrd.g	⊢ (𝜑 → 𝐺 ∈ 𝑊)
mptmpoopabbrd.x	⊢ (𝜑 → 𝑋 ∈ (𝐴‘𝐺))
mptmpoopabbrd.y	⊢ (𝜑 → 𝑌 ∈ (𝐵‘𝐺))
mptmpoopabovd.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
mptmpoopabovd	⊢ (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,𝑓,𝑔,ℎ   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)   𝐴(𝑓, ℎ)   𝐵(𝑓, ℎ)   𝐶(𝑓, ℎ)   𝑀(𝑓, 𝑔, ℎ, 𝑎, 𝑏)   𝑊(𝑓, ℎ, 𝑎, 𝑏)

Proof of Theorem mptmpoopabovd

Step	Hyp	Ref	Expression
1		mptmpoopabbrd.g	. 2 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
2		mptmpoopabbrd.x	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝐴‘𝐺))
3		mptmpoopabbrd.y	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝐵‘𝐺))
4		oveq12 7432	. . 3 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝑎(𝐶‘𝐺)𝑏) = (𝑋(𝐶‘𝐺)𝑌))
5	4	breqd 5125	. 2 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝑓(𝑎(𝐶‘𝐺)𝑏)ℎ ↔ 𝑓(𝑋(𝐶‘𝐺)𝑌)ℎ))
6		fveq2 6888	. . . 4 ⊢ (𝑔 = 𝐺 → (𝐶‘𝑔) = (𝐶‘𝐺))
7	6	oveqd 7440	. . 3 ⊢ (𝑔 = 𝐺 → (𝑎(𝐶‘𝑔)𝑏) = (𝑎(𝐶‘𝐺)𝑏))
8	7	breqd 5125	. 2 ⊢ (𝑔 = 𝐺 → (𝑓(𝑎(𝐶‘𝑔)𝑏)ℎ ↔ 𝑓(𝑎(𝐶‘𝐺)𝑏)ℎ))
9		mptmpoopabovd.m	. 2 ⊢ 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {⟨𝑓, ℎ⟩ ∣ (𝑓(𝑎(𝐶‘𝑔)𝑏)ℎ ∧ 𝑓(𝐷‘𝑔)ℎ)}))
10	1, 2, 3, 5, 8, 9	mptmpoopabbrd 8087	1 ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝑓(𝑋(𝐶‘𝐺)𝑌)ℎ ∧ 𝑓(𝐷‘𝐺)ℎ)})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3458   class class class wbr 5114  {copab 5178   ↦ cmpt 5197  ‘cfv 6543  (class class class)co 7423   ∈ cmpo 7425

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996

This theorem is used by:  wksonproplem  30089  trlsonfval  30090  pthsonfval  30126  spthson  30127