Theorem uprcl4 48863

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⟩(𝐷UP𝐸)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⟩(𝐷UP𝐸)𝑊)𝑀)
2		uprcl4.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
3		eqid 2737	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
4		eqid 2737	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
5		eqid 2737	. . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
6		eqid 2737	. . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸)
7	1, 3	uprcl3 48862	. . . 4 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐸))
8	1	uprcl2 48861	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
9	2, 3, 4, 5, 6, 7, 8	isuplem 48857	. . 3 ⊢ (𝜑 → (𝑋(⟨𝐹, 𝐺⟩(𝐷UP𝐸)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 768	1 ⊢ (𝜑 → 𝑋 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3061  ∃!wreu 3378  ⟨cop 4640   class class class wbr 5151  ‘cfv 6569  (class class class)co 7438  Basecbs 17254  Hom chom 17318  compcco 17319  UPcup 48851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5288  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-iun 5001  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-f1 6574  df-fo 6575  df-f1o 6576  df-fv 6577  df-ov 7441  df-oprab 7442  df-mpo 7443  df-1st 8022  df-2nd 8023  df-func 17918  df-up 48852

This theorem is referenced by:  isup2  48865  upeu3  48866  upeu4  48867