Theorem upeu3 50526

Description: The universal pair ⟨𝑋, 𝑀⟩ from object 𝑊 to functor ⟨𝐹, 𝐺⟩ is essentially unique (strong form) if it exists. (Contributed by Zhi Wang, 24-Sep-2025.)

Hypotheses

Ref	Expression
upeu3.i	⊢ (𝜑 → 𝐼 = (Iso‘𝐷))
upeu3.o	⊢ (𝜑 → ⚬ = (⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌)))
upeu3.x	⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀)
upeu3.y	⊢ (𝜑 → 𝑌(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑁)

Assertion

Ref	Expression
upeu3	⊢ (𝜑 → ∃!𝑟 ∈ (𝑋𝐼𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟) ⚬ 𝑀))

Distinct variable groups:   𝐷,𝑟   𝐸,𝑟   𝐹,𝑟   𝐺,𝑟   𝑀,𝑟   𝑁,𝑟   𝑊,𝑟   𝑋,𝑟   𝑌,𝑟   𝜑,𝑟

Allowed substitution hints:   𝐼(𝑟)   ⚬ (𝑟)

Proof of Theorem upeu3

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

Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
2		eqid 2763	. . 3 ⊢ (Base‘𝐸) = (Base‘𝐸)
3		eqid 2763	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		eqid 2763	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
5		eqid 2763	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
6		upeu3.x	. . . 4 ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀)
7	6	uprcl2 50520	. . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
8	6, 1	uprcl4 50522	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))
9		upeu3.y	. . . 4 ⊢ (𝜑 → 𝑌(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑁)
10	9, 1	uprcl4 50522	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
11	6, 2	uprcl3 50521	. . 3 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐸))
12	6, 4	uprcl5 50523	. . 3 ⊢ (𝜑 → 𝑀 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑋)))
13	1, 3, 4, 5, 6	isup2 50525	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝐷)∀𝑔 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑦))∃!𝑘 ∈ (𝑋(Hom ‘𝐷)𝑦)𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑦))𝑀))
14	9, 4	uprcl5 50523	. . 3 ⊢ (𝜑 → 𝑁 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑌)))
15	1, 3, 4, 5, 9	isup2 50525	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝐷)∀𝑔 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑦))∃!𝑘 ∈ (𝑌(Hom ‘𝐷)𝑦)𝑔 = (((𝑌𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑌)⟩(comp‘𝐸)(𝐹‘𝑦))𝑁))
16	1, 2, 3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 15	upeu 50502	. 2 ⊢ (𝜑 → ∃!𝑟 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌))𝑀))
17		upeu3.i	. . . 4 ⊢ (𝜑 → 𝐼 = (Iso‘𝐷))
18	17	oveqd 7463	. . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = (𝑋(Iso‘𝐷)𝑌))
19		upeu3.o	. . . . 5 ⊢ (𝜑 → ⚬ = (⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌)))
20	19	oveqd 7463	. . . 4 ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝑟) ⚬ 𝑀) = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌))𝑀))
21	20	eqeq2d 2774	. . 3 ⊢ (𝜑 → (𝑁 = (((𝑋𝐺𝑌)‘𝑟) ⚬ 𝑀) ↔ 𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌))𝑀)))
22	18, 21	reueqbidv 3405	. 2 ⊢ (𝜑 → (∃!𝑟 ∈ (𝑋𝐼𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟) ⚬ 𝑀) ↔ ∃!𝑟 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌))𝑀)))
23	16, 22	mpbird 260	1 ⊢ (𝜑 → ∃!𝑟 ∈ (𝑋𝐼𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟) ⚬ 𝑀))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ∃!wreu 3367  ⟨cop 4838   class class class wbr 5106  ‘cfv 6561  (class class class)co 7446  Basecbs 17419  Hom chom 17471  compcco 17472  Isociso 17953   UP cup 50504

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 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

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-rmo 3369  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 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-supp 8208  df-map 8880  df-ixp 8957  df-cat 17874  df-cid 17875  df-sect 17954  df-inv 17955  df-iso 17956  df-cic 18003  df-func 18065  df-up 50505

This theorem is used by: (None)