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	⊢ (𝜑 → 𝐺 ∈ 𝑊)
mptmpoopabbrd.x	⊢ (𝜑 → 𝑋 ∈ (𝐴‘𝐺))
mptmpoopabbrd.y	⊢ (𝜑 → 𝑌 ∈ (𝐵‘𝐺))
mptmpoopabbrd.v	⊢ (𝜑 → {⟨𝑓, ℎ⟩ ∣ 𝜓} ∈ 𝑉)
mptmpoopabbrd.r	⊢ ((𝜑 ∧ 𝑓(𝐷‘𝐺)ℎ) → 𝜓)
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,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 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		mptmpoopabbrd.v	. 2 ⊢ (𝜑 → {⟨𝑓, ℎ⟩ ∣ 𝜓} ∈ 𝑉)
5		mptmpoopabbrd.r	. 2 ⊢ ((𝜑 ∧ 𝑓(𝐷‘𝐺)ℎ) → 𝜓)
6		oveq12 7159	. . 3 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝑎(𝐶‘𝐺)𝑏) = (𝑋(𝐶‘𝐺)𝑌))
7	6	breqd 5043	. 2 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝑓(𝑎(𝐶‘𝐺)𝑏)ℎ ↔ 𝑓(𝑋(𝐶‘𝐺)𝑌)ℎ))
8		fveq2 6658	. . . 4 ⊢ (𝑔 = 𝐺 → (𝐶‘𝑔) = (𝐶‘𝐺))
9	8	oveqd 7167	. . 3 ⊢ (𝑔 = 𝐺 → (𝑎(𝐶‘𝑔)𝑏) = (𝑎(𝐶‘𝐺)𝑏))
10	9	breqd 5043	. 2 ⊢ (𝑔 = 𝐺 → (𝑓(𝑎(𝐶‘𝑔)𝑏)ℎ ↔ 𝑓(𝑎(𝐶‘𝐺)𝑏)ℎ))
11		mptmpoopabovd.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	mptmpoopabbrd 7783	1 ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝑓(𝑋(𝐶‘𝐺)𝑌)ℎ ∧ 𝑓(𝐷‘𝐺)ℎ)})

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