Users' Mathboxes Mathbox for Rohan Ridenour < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mnuop23d Structured version   Visualization version   GIF version

Theorem mnuop23d 43583
Description: Second and third operations of a minimal universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Hypotheses
Ref Expression
mnuop23d.1 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
mnuop23d.2 (𝜑𝑈𝑀)
mnuop23d.3 (𝜑𝐴𝑈)
mnuop23d.4 (𝜑𝐹𝑉)
Assertion
Ref Expression
mnuop23d (𝜑 → ∃𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝐹) → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))))
Distinct variable groups:   𝑣,𝐹   𝑤,𝐴,𝑖   𝑤,𝑢,𝐹,𝑖   𝑤,𝑣,𝑈,𝑘,𝑚,𝑛,𝑞,𝑝,𝑙,𝑖   𝑤,𝑟   𝑢,𝑈,𝑘,𝑚,𝑛,𝑟,𝑝,𝑙,𝑖
Allowed substitution hints:   𝜑(𝑤,𝑣,𝑢,𝑖,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐴(𝑣,𝑢,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐹(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝑀(𝑤,𝑣,𝑢,𝑖,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝑉(𝑤,𝑣,𝑢,𝑖,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem mnuop23d
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 mnuop23d.4 . 2 (𝜑𝐹𝑉)
2 mnuop23d.1 . . . 4 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
3 mnuop23d.2 . . . 4 (𝜑𝑈𝑀)
4 mnuop23d.3 . . . 4 (𝜑𝐴𝑈)
52, 3, 4mnuop123d 43579 . . 3 (𝜑 → (𝒫 𝐴𝑈 ∧ ∀𝑓𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)))))
65simprd 495 . 2 (𝜑 → ∀𝑓𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤))))
7 eleq2 2816 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑣𝑓𝑣𝐹))
87anbi2d 628 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑖𝑣𝑣𝑓) ↔ (𝑖𝑣𝑣𝐹)))
98rexbidv 3172 . . . . . . 7 (𝑓 = 𝐹 → (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) ↔ ∃𝑣𝑈 (𝑖𝑣𝑣𝐹)))
10 rexeq 3315 . . . . . . 7 (𝑓 = 𝐹 → (∃𝑢𝑓 (𝑖𝑢 𝑢𝑤) ↔ ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤)))
119, 10imbi12d 344 . . . . . 6 (𝑓 = 𝐹 → ((∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)) ↔ (∃𝑣𝑈 (𝑖𝑣𝑣𝐹) → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))))
1211ralbidv 3171 . . . . 5 (𝑓 = 𝐹 → (∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)) ↔ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝐹) → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))))
1312anbi2d 628 . . . 4 (𝑓 = 𝐹 → ((𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤))) ↔ (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝐹) → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤)))))
1413rexbidv 3172 . . 3 (𝑓 = 𝐹 → (∃𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤))) ↔ ∃𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝐹) → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤)))))
1514spcgv 3580 . 2 (𝐹𝑉 → (∀𝑓𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤))) → ∃𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝐹) → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤)))))
161, 6, 15sylc 65 1 (𝜑 → ∃𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝐹) → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1531   = wceq 1533  wcel 2098  {cab 2703  wral 3055  wrex 3064  wss 3943  𝒫 cpw 4597   cuni 4902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-v 3470  df-in 3950  df-ss 3960  df-pw 4599  df-uni 4903
This theorem is referenced by:  mnupwd  43584  mnuop3d  43588
  Copyright terms: Public domain W3C validator