Theorem upeu4 49185

Description: Generate a new universal morphism through an isomorphism from an existing universal object, and pair with the codomain of the isomorphism to form a universal pair. (Contributed by Zhi Wang, 25-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)
upeu4.k	⊢ (𝜑 → 𝐾 ∈ (𝑋𝐼𝑌))
upeu4.n	⊢ (𝜑 → 𝑁 = (((𝑋𝐺𝑌)‘𝐾) ⚬ 𝑀))

Assertion

Ref	Expression
upeu4	⊢ (𝜑 → 𝑌(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑁)

Proof of Theorem upeu4

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

Step	Hyp	Ref	Expression
1		eqid 2729	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
2		eqid 2729	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
3		eqid 2729	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		eqid 2729	. . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
5		eqid 2729	. . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸)
6		upeu3.x	. . . . 5 ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀)
7	6	uprcl2 49178	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
8	6, 1	uprcl4 49180	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))
9		upeu4.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐼𝑌))
10	7	funcrcl2 49068	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ Cat)
11		isofn 17737	. . . . . . . . . 10 ⊢ (𝐷 ∈ Cat → (Iso‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷)))
12	10, 11	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (Iso‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷)))
13		upeu3.i	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 = (Iso‘𝐷))
14	13	fneq1d 6611	. . . . . . . . 9 ⊢ (𝜑 → (𝐼 Fn ((Base‘𝐷) × (Base‘𝐷)) ↔ (Iso‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷))))
15	12, 14	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → 𝐼 Fn ((Base‘𝐷) × (Base‘𝐷)))
16		fnov 7520	. . . . . . . 8 ⊢ (𝐼 Fn ((Base‘𝐷) × (Base‘𝐷)) ↔ 𝐼 = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐼𝑦)))
17	15, 16	sylib 218	. . . . . . 7 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐼𝑦)))
18	17	oveqd 7404	. . . . . 6 ⊢ (𝜑 → (𝑋𝐼𝑌) = (𝑋(𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐼𝑦))𝑌))
19	9, 18	eleqtrd 2830	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝑋(𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐼𝑦))𝑌))
20		eqid 2729	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐼𝑦)) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐼𝑦))
21	20	elmpocl2 7632	. . . . 5 ⊢ (𝐾 ∈ (𝑋(𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐼𝑦))𝑌) → 𝑌 ∈ (Base‘𝐷))
22	19, 21	syl 17	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
23	6, 2	uprcl3 49179	. . . 4 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐸))
24	6, 4	uprcl5 49181	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑋)))
25	1, 3, 4, 5, 6	isup2 49183	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐷)∀𝑓 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑥))∃!𝑘 ∈ (𝑋(Hom ‘𝐷)𝑥)𝑓 = (((𝑋𝐺𝑥)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑥))𝑀))
26		eqid 2729	. . . 4 ⊢ (Iso‘𝐷) = (Iso‘𝐷)
27	13	oveqd 7404	. . . . 5 ⊢ (𝜑 → (𝑋𝐼𝑌) = (𝑋(Iso‘𝐷)𝑌))
28	9, 27	eleqtrd 2830	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝑋(Iso‘𝐷)𝑌))
29		upeu4.n	. . . . 5 ⊢ (𝜑 → 𝑁 = (((𝑋𝐺𝑌)‘𝐾) ⚬ 𝑀))
30		upeu3.o	. . . . . 6 ⊢ (𝜑 → ⚬ = (⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌)))
31	30	oveqd 7404	. . . . 5 ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝐾) ⚬ 𝑀) = (((𝑋𝐺𝑌)‘𝐾)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌))𝑀))
32	29, 31	eqtrd 2764	. . . 4 ⊢ (𝜑 → 𝑁 = (((𝑋𝐺𝑌)‘𝐾)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌))𝑀))
33	1, 2, 3, 4, 5, 7, 8, 22, 23, 24, 25, 26, 28, 32	upeu2 49161	. . 3 ⊢ (𝜑 → (𝑁 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑌)) ∧ ∀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)))
34	33	simprd 495	. 2 ⊢ (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))
35	33	simpld 494	. . 3 ⊢ (𝜑 → 𝑁 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑌)))
36	1, 2, 3, 4, 5, 23, 7, 22, 35	isup 49169	. 2 ⊢ (𝜑 → (𝑌(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑁 ↔ ∀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)))
37	34, 36	mpbird 257	1 ⊢ (𝜑 → 𝑌(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑁)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃!wreu 3352  ⟨cop 4595   class class class wbr 5107   × cxp 5636   Fn wfn 6506  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  Basecbs 17179  Hom chom 17231  compcco 17232  Catccat 17625  Isociso 17708   UP cup 49162

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-map 8801  df-ixp 8871  df-cat 17629  df-cid 17630  df-sect 17709  df-inv 17710  df-iso 17711  df-func 17820  df-up 49163

This theorem is referenced by: (None)