Theorem uprcl4 49544

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 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 49543	. . . 4 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐸))
8	1	uprcl2 49542	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
9	2, 3, 4, 5, 6, 7, 8	isuplem 49532	. . 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 768	1 ⊢ (𝜑 → 𝑋 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃!wreu 3350  ⟨cop 4588   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  Hom chom 17200  compcco 17201   UP cup 49526

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-func 17794  df-up 49527

This theorem is referenced by:  isup2  49547  upeu3  49548  upeu4  49549  oppcuprcl4  49552  uptr  49566  uptrar  49569  isinito2  49852  isinito3  49853  lanrcl4  49987  iscmd  50019  cmdlan  50025