Theorem upeu3 50247

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 2761	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
2		eqid 2761	. . 3 ⊢ (Base‘𝐸) = (Base‘𝐸)
3		eqid 2761	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		eqid 2761	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
5		eqid 2761	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
6		upeu3.x	. . . 4 ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀)
7	6	uprcl2 50241	. . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
8	6, 1	uprcl4 50243	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))
9		upeu3.y	. . . 4 ⊢ (𝜑 → 𝑌(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑁)
10	9, 1	uprcl4 50243	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
11	6, 2	uprcl3 50242	. . 3 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐸))
12	6, 4	uprcl5 50244	. . 3 ⊢ (𝜑 → 𝑀 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑋)))
13	1, 3, 4, 5, 6	isup2 50246	. . 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 50244	. . 3 ⊢ (𝜑 → 𝑁 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑌)))
15	1, 3, 4, 5, 9	isup2 50246	. . 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 50223	. 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 7429	. . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = (𝑋(Iso‘𝐷)𝑌))
19		upeu3.o	. . . . 5 ⊢ (𝜑 → ⚬ = (⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌)))
20	19	oveqd 7429	. . . 4 ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝑟) ⚬ 𝑀) = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌))𝑀))
21	20	eqeq2d 2772	. . 3 ⊢ (𝜑 → (𝑁 = (((𝑋𝐺𝑌)‘𝑟) ⚬ 𝑀) ↔ 𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌))𝑀)))
22	18, 21	reueqbidv 3402	. 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 3364  ⟨cop 4590   class class class wbr 5103  ‘cfv 6531  (class class class)co 7412  Basecbs 17367  Hom chom 17419  compcco 17420  Isociso 17901   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-rmo 3366  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-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-supp 8162  df-map 8833  df-ixp 8910  df-cat 17822  df-cid 17823  df-sect 17902  df-inv 17903  df-iso 17904  df-cic 17951  df-func 18013  df-up 50226

This theorem is used by: (None)