Theorem uprcl4 49222

Description: Reverse closure for the class of universal property. (Contributed by Zhi Wang, 25-Sep-2025.)

Hypotheses

Ref	Expression
uprcl2.x	⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀)
uprcl4.b	⊢ 𝐵 = (Base‘𝐷)

Assertion

Ref	Expression
uprcl4	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Proof of Theorem uprcl4

Dummy variables 𝑔 𝑘 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uprcl2.x	. . 3 ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀)
2		uprcl4.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
3		eqid 2731	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
4		eqid 2731	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
5		eqid 2731	. . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
6		eqid 2731	. . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸)
7	1, 3	uprcl3 49221	. . . 4 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐸))
8	1	uprcl2 49220	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
9	2, 3, 4, 5, 6, 7, 8	isuplem 49210	. . 3 ⊢ (𝜑 → (𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀 ↔ ((𝑋 ∈ 𝐵 ∧ 𝑀 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑋))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑦))∃!𝑘 ∈ (𝑋(Hom ‘𝐷)𝑦)𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑦))𝑀))))
10	1, 9	mpbid 232	. 2 ⊢ (𝜑 → ((𝑋 ∈ 𝐵 ∧ 𝑀 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑋))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑦))∃!𝑘 ∈ (𝑋(Hom ‘𝐷)𝑦)𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑦))𝑀)))
11	10	simplld 767	1 ⊢ (𝜑 → 𝑋 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃!wreu 3344  ⟨cop 4582   class class class wbr 5091  ‘cfv 6481  (class class class)co 7346  Basecbs 17117  Hom chom 17169  compcco 17170   UP cup 49204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-func 17762  df-up 49205

This theorem is referenced by:  isup2  49225  upeu3  49226  upeu4  49227  oppcuprcl4  49230  uptr  49244  uptrar  49247  isinito2  49530  isinito3  49531  lanrcl4  49665  iscmd  49697  cmdlan  49703