Theorem upeu3 48866

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⟩(𝐷UP𝐸)𝑊)𝑀)
upeu3.y	⊢ (𝜑 → 𝑌(⟨𝐹, 𝐺⟩(𝐷UP𝐸)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 2737	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
2		eqid 2737	. . 3 ⊢ (Base‘𝐸) = (Base‘𝐸)
3		eqid 2737	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		eqid 2737	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
5		eqid 2737	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
6		upeu3.x	. . . 4 ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷UP𝐸)𝑊)𝑀)
7	6	uprcl2 48861	. . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
8	6, 1	uprcl4 48863	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))
9		upeu3.y	. . . 4 ⊢ (𝜑 → 𝑌(⟨𝐹, 𝐺⟩(𝐷UP𝐸)𝑊)𝑁)
10	9, 1	uprcl4 48863	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
11	6, 2	uprcl3 48862	. . 3 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐸))
12	6, 4	uprcl5 48864	. . 3 ⊢ (𝜑 → 𝑀 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑋)))
13	1, 3, 4, 5, 6	isup2 48865	. . 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 48864	. . 3 ⊢ (𝜑 → 𝑁 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑌)))
15	1, 3, 4, 5, 9	isup2 48865	. . 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 48849	. 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 7455	. . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = (𝑋(Iso‘𝐷)𝑌))
19		upeu3.o	. . . . 5 ⊢ (𝜑 → ⚬ = (⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌)))
20	19	oveqd 7455	. . . 4 ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝑟) ⚬ 𝑀) = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌))𝑀))
21	20	eqeq2d 2748	. . 3 ⊢ (𝜑 → (𝑁 = (((𝑋𝐺𝑌)‘𝑟) ⚬ 𝑀) ↔ 𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌))𝑀)))
22	18, 21	reueqbidv 3423	. 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 1539  ∃!wreu 3378  ⟨cop 4640   class class class wbr 5151  ‘cfv 6569  (class class class)co 7438  Basecbs 17254  Hom chom 17318  compcco 17319  Isociso 17803  UPcup 48851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5288  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-iun 5001  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-f1 6574  df-fo 6575  df-f1o 6576  df-fv 6577  df-riota 7395  df-ov 7441  df-oprab 7442  df-mpo 7443  df-1st 8022  df-2nd 8023  df-supp 8194  df-map 8876  df-ixp 8946  df-cat 17722  df-cid 17723  df-sect 17804  df-inv 17805  df-iso 17806  df-cic 17853  df-func 17918  df-up 48852

This theorem is referenced by: (None)