Theorem upeu3 49174

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 2730	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
2		eqid 2730	. . 3 ⊢ (Base‘𝐸) = (Base‘𝐸)
3		eqid 2730	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		eqid 2730	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
5		eqid 2730	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
6		upeu3.x	. . . 4 ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀)
7	6	uprcl2 49168	. . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
8	6, 1	uprcl4 49170	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))
9		upeu3.y	. . . 4 ⊢ (𝜑 → 𝑌(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑁)
10	9, 1	uprcl4 49170	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
11	6, 2	uprcl3 49169	. . 3 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐸))
12	6, 4	uprcl5 49171	. . 3 ⊢ (𝜑 → 𝑀 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑋)))
13	1, 3, 4, 5, 6	isup2 49173	. . 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 49171	. . 3 ⊢ (𝜑 → 𝑁 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑌)))
15	1, 3, 4, 5, 9	isup2 49173	. . 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 49150	. 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 7406	. . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = (𝑋(Iso‘𝐷)𝑌))
19		upeu3.o	. . . . 5 ⊢ (𝜑 → ⚬ = (⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌)))
20	19	oveqd 7406	. . . 4 ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝑟) ⚬ 𝑀) = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌))𝑀))
21	20	eqeq2d 2741	. . 3 ⊢ (𝜑 → (𝑁 = (((𝑋𝐺𝑌)‘𝑟) ⚬ 𝑀) ↔ 𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌))𝑀)))
22	18, 21	reueqbidv 3397	. 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 257	1 ⊢ (𝜑 → ∃!𝑟 ∈ (𝑋𝐼𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟) ⚬ 𝑀))

Syntax hints:   → wi 4   = wceq 1540  ∃!wreu 3354  ⟨cop 4597   class class class wbr 5109  ‘cfv 6513  (class class class)co 7389  Basecbs 17185  Hom chom 17237  compcco 17238  Isociso 17714   UP cup 49152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-1st 7970  df-2nd 7971  df-supp 8142  df-map 8803  df-ixp 8873  df-cat 17635  df-cid 17636  df-sect 17715  df-inv 17716  df-iso 17717  df-cic 17764  df-func 17826  df-up 49153

This theorem is referenced by: (None)