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Theorem mnuop23d 40894
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 40890 . . 3 (𝜑 → (𝒫 𝐴𝑈 ∧ ∀𝑓𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)))))
65simprd 499 . 2 (𝜑 → ∀𝑓𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤))))
7 eleq2 2904 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑣𝑓𝑣𝐹))
87anbi2d 631 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑖𝑣𝑣𝑓) ↔ (𝑖𝑣𝑣𝐹)))
98rexbidv 3289 . . . . . . 7 (𝑓 = 𝐹 → (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) ↔ ∃𝑣𝑈 (𝑖𝑣𝑣𝐹)))
10 rexeq 3397 . . . . . . 7 (𝑓 = 𝐹 → (∃𝑢𝑓 (𝑖𝑢 𝑢𝑤) ↔ ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤)))
119, 10imbi12d 348 . . . . . 6 (𝑓 = 𝐹 → ((∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)) ↔ (∃𝑣𝑈 (𝑖𝑣𝑣𝐹) → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))))
1211ralbidv 3192 . . . . 5 (𝑓 = 𝐹 → (∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)) ↔ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝐹) → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))))
1312anbi2d 631 . . . 4 (𝑓 = 𝐹 → ((𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤))) ↔ (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝐹) → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤)))))
1413rexbidv 3289 . . 3 (𝑓 = 𝐹 → (∃𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤))) ↔ ∃𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝐹) → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤)))))
1514spcgv 3581 . 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 399  wal 1536   = wceq 1538  wcel 2115  {cab 2802  wral 3133  wrex 3134  wss 3919  𝒫 cpw 4522   cuni 4824
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-in 3926  df-ss 3936  df-pw 4524  df-uni 4825
This theorem is referenced by:  mnupwd  40895  mnuop3d  40899
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