Theorem uprcl5 50244

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)
uprcl5.j	⊢ 𝐽 = (Hom ‘𝐸)

Assertion

Ref	Expression
uprcl5	⊢ (𝜑 → 𝑀 ∈ (𝑊𝐽(𝐹‘𝑋)))

Proof of Theorem uprcl5

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		eqid 2761	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		eqid 2761	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
4		eqid 2761	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
5		uprcl5.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐸)
6		eqid 2761	. . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸)
7	1, 3	uprcl3 50242	. . . 4 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐸))
8	1	uprcl2 50241	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
9	2, 3, 4, 5, 6, 7, 8	isuplem 50231	. . 3 ⊢ (𝜑 → (𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀 ↔ ((𝑋 ∈ (Base‘𝐷) ∧ 𝑀 ∈ (𝑊𝐽(𝐹‘𝑋))) ∧ ∀𝑦 ∈ (Base‘𝐷)∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑋(Hom ‘𝐷)𝑦)𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑦))𝑀))))
10	1, 9	mpbid 235	. 2 ⊢ (𝜑 → ((𝑋 ∈ (Base‘𝐷) ∧ 𝑀 ∈ (𝑊𝐽(𝐹‘𝑋))) ∧ ∀𝑦 ∈ (Base‘𝐷)∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑋(Hom ‘𝐷)𝑦)𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑦))𝑀)))
11	10	simplrd 782	1 ⊢ (𝜑 → 𝑀 ∈ (𝑊𝐽(𝐹‘𝑋)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃!wreu 3364  ⟨cop 4590   class class class wbr 5103  ‘cfv 6531  (class class class)co 7412  Basecbs 17367  Hom chom 17419  compcco 17420   UP cup 50225

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-func 18013  df-up 50226

This theorem is used by:  isup2  50246  upeu3  50247  upeu4  50248  oppcuprcl5  50253  uptri  50266  uptrar  50268  uptrai  50269  uptr2  50273  isinito3  50552  lanrcl5  50687  iscmd  50718  lmddu  50719