Theorem upeu4 49437

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 2736	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
2		eqid 2736	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
3		eqid 2736	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		eqid 2736	. . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
5		eqid 2736	. . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸)
6		upeu3.x	. . . . 5 ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀)
7	6	uprcl2 49430	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
8	6, 1	uprcl4 49432	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))
9		upeu4.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐼𝑌))
10	7	funcrcl2 49320	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ Cat)
11		isofn 17699	. . . . . . . . . 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 6585	. . . . . . . . 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 7489	. . . . . . . 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 7375	. . . . . 6 ⊢ (𝜑 → (𝑋𝐼𝑌) = (𝑋(𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐼𝑦))𝑌))
19	9, 18	eleqtrd 2838	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝑋(𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐼𝑦))𝑌))
20		eqid 2736	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐼𝑦)) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐼𝑦))
21	20	elmpocl2 7601	. . . . 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 49431	. . . 4 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐸))
24	6, 4	uprcl5 49433	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑊(Hom ‘𝐸)(𝐹‘𝑋)))
25	1, 3, 4, 5, 6	isup2 49435	. . . 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 2736	. . . 4 ⊢ (Iso‘𝐷) = (Iso‘𝐷)
27	13	oveqd 7375	. . . . 5 ⊢ (𝜑 → (𝑋𝐼𝑌) = (𝑋(Iso‘𝐷)𝑌))
28	9, 27	eleqtrd 2838	. . . 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 7375	. . . . 5 ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝐾) ⚬ 𝑀) = (((𝑋𝐺𝑌)‘𝐾)(⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌))𝑀))
32	29, 31	eqtrd 2771	. . . 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 49413	. . 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 49421	. 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 1541   ∈ wcel 2113  ∀wral 3051  ∃!wreu 3348  ⟨cop 4586   class class class wbr 5098   × cxp 5622   Fn wfn 6487  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  Basecbs 17136  Hom chom 17188  compcco 17189  Catccat 17587  Isociso 17670   UP cup 49414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8765  df-ixp 8836  df-cat 17591  df-cid 17592  df-sect 17671  df-inv 17672  df-iso 17673  df-func 17782  df-up 49415

This theorem is referenced by: (None)