Theorem uprcl5 50513

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 2763	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		eqid 2763	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
4		eqid 2763	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
5		uprcl5.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐸)
6		eqid 2763	. . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸)
7	1, 3	uprcl3 50511	. . . 4 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐸))
8	1	uprcl2 50510	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
9	2, 3, 4, 5, 6, 7, 8	isuplem 50500	. . 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 2147  ∀wral 3080  ∃!wreu 3367  ⟨cop 4838   class class class wbr 5106  ‘cfv 6558  (class class class)co 7443  Basecbs 17416  Hom chom 17468  compcco 17469   UP cup 50494

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-func 18062  df-up 50495

This theorem is used by:  isup2  50515  upeu3  50516  upeu4  50517  oppcuprcl5  50522  uptri  50535  uptrar  50537  uptrai  50538  uptr2  50542  isinito3  50840  lanrcl5  51023  iscmd  51054  lmddu  51055