Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  mptmpt2opabovd Structured version   Visualization version   GIF version

Theorem mptmpt2opabovd 7485
 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
mptmpt2opabbrd.g (𝜑𝐺𝑊)
mptmpt2opabbrd.x (𝜑𝑋 ∈ (𝐴𝐺))
mptmpt2opabbrd.y (𝜑𝑌 ∈ (𝐵𝐺))
mptmpt2opabbrd.v (𝜑 → {⟨𝑓, ⟩ ∣ 𝜓} ∈ 𝑉)
mptmpt2opabbrd.r ((𝜑𝑓(𝐷𝐺)) → 𝜓)
mptmpt2opabovd.m 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝑓(𝑎(𝐶𝑔)𝑏)𝑓(𝐷𝑔))}))
Assertion
Ref Expression
mptmpt2opabovd (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 mptmpt2opabovd
StepHypRef Expression
1 mptmpt2opabbrd.g . 2 (𝜑𝐺𝑊)
2 mptmpt2opabbrd.x . 2 (𝜑𝑋 ∈ (𝐴𝐺))
3 mptmpt2opabbrd.y . 2 (𝜑𝑌 ∈ (𝐵𝐺))
4 mptmpt2opabbrd.v . 2 (𝜑 → {⟨𝑓, ⟩ ∣ 𝜓} ∈ 𝑉)
5 mptmpt2opabbrd.r . 2 ((𝜑𝑓(𝐷𝐺)) → 𝜓)
6 oveq12 6887 . . 3 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝑎(𝐶𝐺)𝑏) = (𝑋(𝐶𝐺)𝑌))
76breqd 4854 . 2 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝑓(𝑎(𝐶𝐺)𝑏)𝑓(𝑋(𝐶𝐺)𝑌)))
8 fveq2 6411 . . . 4 (𝑔 = 𝐺 → (𝐶𝑔) = (𝐶𝐺))
98oveqd 6895 . . 3 (𝑔 = 𝐺 → (𝑎(𝐶𝑔)𝑏) = (𝑎(𝐶𝐺)𝑏))
109breqd 4854 . 2 (𝑔 = 𝐺 → (𝑓(𝑎(𝐶𝑔)𝑏)𝑓(𝑎(𝐶𝐺)𝑏)))
11 mptmpt2opabovd.m . 2 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝑓(𝑎(𝐶𝑔)𝑏)𝑓(𝐷𝑔))}))
121, 2, 3, 4, 5, 7, 10, 11mptmpt2opabbrd 7484 1 (𝜑 → (𝑋(𝑀𝐺)𝑌) = {⟨𝑓, ⟩ ∣ (𝑓(𝑋(𝐶𝐺)𝑌)𝑓(𝐷𝐺))})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157  Vcvv 3385   class class class wbr 4843  {copab 4905   ↦ cmpt 4922  ‘cfv 6101  (class class class)co 6878   ↦ cmpt2 6880 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402 This theorem is referenced by:  wksonproplem  26959  trlsonfval  26960  pthsonfval  26994  spthson  26995
 Copyright terms: Public domain W3C validator