Theorem upeu4 49825

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 2765	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
2		eqid 2765	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
3		eqid 2765	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		eqid 2765	. . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
5		eqid 2765	. . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸)
6		upeu3.x	. . . . 5 ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀)
7	6	uprcl2 49818	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
8	6, 1	uprcl4 49820	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))
9		upeu4.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐼𝑌))
10	7	funcrcl2 49708	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ Cat)
11		isofn 17822	. . . . . . . . . 10 ⊢ (𝐷 ∈ Cat → (Iso‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷)))
12	10, 11	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (Iso‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷)))
13		upeu3.i	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 = (Iso‘𝐷))
14	13	fneq1d 6618	. . . . . . . . 9 ⊢ (𝜑 → (𝐼 Fn ((Base‘𝐷) × (Base‘𝐷)) ↔ (Iso‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷))))
15	12, 14	mpbird 260	. . . . . . . 8 ⊢ (𝜑 → 𝐼 Fn ((Base‘𝐷) × (Base‘𝐷)))
16		fnov 7531	. . . . . . . 8 ⊢ (𝐼 Fn ((Base‘𝐷) × (Base‘𝐷)) ↔ 𝐼 = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐼𝑦)))
17	15, 16	sylib 221	. . . . . . 7 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐼𝑦)))
18	17	oveqd 7417	. . . . . 6 ⊢ (𝜑 → (𝑋𝐼𝑌) = (𝑋(𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐼𝑦))𝑌))
19	9, 18	eleqtrd 2867	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝑋(𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐼𝑦))𝑌))
20		eqid 2765	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐼𝑦)) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐼𝑦))
21	20	elmpocl2 7643	. . . . 5 ⊢ (𝐾 ∈ (𝑋(𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐼𝑦))𝑌) → 𝑌 ∈ (Base‘𝐷))
22	19, 21	syl 18	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
23	6, 2	uprcl3 49819	. . . 4 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐸))
24	6, 4	uprcl5 49821	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑋)))
25	1, 3, 4, 5, 6	isup2 49823	. . . 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 2765	. . . 4 ⊢ (Iso‘𝐷) = (Iso‘𝐷)
27	13	oveqd 7417	. . . . 5 ⊢ (𝜑 → (𝑋𝐼𝑌) = (𝑋(Iso‘𝐷)𝑌))
28	9, 27	eleqtrd 2867	. . . 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 7417	. . . . 5 ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝐾) ⚬ 𝑀) = (((𝑋𝐺𝑌)‘𝐾)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌))𝑀))
32	29, 31	eqtrd 2800	. . . 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 49801	. . 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 500	. 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 499	. . 3 ⊢ (𝜑 → 𝑁 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑌)))
36	1, 2, 3, 4, 5, 23, 7, 22, 35	isup 49809	. 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 260	1 ⊢ (𝜑 → 𝑌(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑁)

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∃!wreu 3368  ⟨cop 4591   class class class wbr 5105   × cxp 5650   Fn wfn 6520  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  Basecbs 17259  Hom chom 17311  compcco 17312  Catccat 17710  Isociso 17793   UP cup 49802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-map 8814  df-ixp 8884  df-cat 17714  df-cid 17715  df-sect 17794  df-inv 17795  df-iso 17796  df-func 17905  df-up 49803

This theorem is referenced by: (None)