Theorem upeu3 50404

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 2760	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
2		eqid 2760	. . 3 ⊢ (Base‘𝐸) = (Base‘𝐸)
3		eqid 2760	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		eqid 2760	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
5		eqid 2760	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
6		upeu3.x	. . . 4 ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀)
7	6	uprcl2 50398	. . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
8	6, 1	uprcl4 50400	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))
9		upeu3.y	. . . 4 ⊢ (𝜑 → 𝑌(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑁)
10	9, 1	uprcl4 50400	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
11	6, 2	uprcl3 50399	. . 3 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐸))
12	6, 4	uprcl5 50401	. . 3 ⊢ (𝜑 → 𝑀 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑋)))
13	1, 3, 4, 5, 6	isup2 50403	. . 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 50401	. . 3 ⊢ (𝜑 → 𝑁 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑌)))
15	1, 3, 4, 5, 9	isup2 50403	. . 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 50380	. 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 7437	. . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = (𝑋(Iso‘𝐷)𝑌))
19		upeu3.o	. . . . 5 ⊢ (𝜑 → ⚬ = (⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌)))
20	19	oveqd 7437	. . . 4 ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝑟) ⚬ 𝑀) = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌))𝑀))
21	20	eqeq2d 2771	. . 3 ⊢ (𝜑 → (𝑁 = (((𝑋𝐺𝑌)‘𝑟) ⚬ 𝑀) ↔ 𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌))𝑀)))
22	18, 21	reueqbidv 3401	. 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 1569  ∃!wreu 3363  ⟨cop 4834   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Basecbs 17392  Hom chom 17444  compcco 17445  Isociso 17926   UP cup 50382

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-supp 8180  df-map 8852  df-ixp 8929  df-cat 17847  df-cid 17848  df-sect 17927  df-inv 17928  df-iso 17929  df-cic 17976  df-func 18038  df-up 50383

This theorem is used by: (None)