Theorem upcic 49828

Description: A universal property defines an object up to isomorphism given its existence. (Contributed by Zhi Wang, 17-Sep-2025.)

Hypotheses

Ref	Expression
upcic.b	⊢ 𝐵 = (Base‘𝐷)
upcic.c	⊢ 𝐶 = (Base‘𝐸)
upcic.h	⊢ 𝐻 = (Hom ‘𝐷)
upcic.j	⊢ 𝐽 = (Hom ‘𝐸)
upcic.o	⊢ 𝑂 = (comp‘𝐸)
upcic.f	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
upcic.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
upcic.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
upcic.z	⊢ (𝜑 → 𝑍 ∈ 𝐶)
upcic.m	⊢ (𝜑 → 𝑀 ∈ (𝑍𝐽(𝐹‘𝑋)))
upcic.1	⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ∀𝑓 ∈ (𝑍𝐽(𝐹‘𝑤))∃!𝑘 ∈ (𝑋𝐻𝑤)𝑓 = (((𝑋𝐺𝑤)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑤))𝑀))
upcic.n	⊢ (𝜑 → 𝑁 ∈ (𝑍𝐽(𝐹‘𝑌)))
upcic.2	⊢ (𝜑 → ∀𝑣 ∈ 𝐵 ∀𝑔 ∈ (𝑍𝐽(𝐹‘𝑣))∃!𝑙 ∈ (𝑌𝐻𝑣)𝑔 = (((𝑌𝐺𝑣)‘𝑙)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑣))𝑁))

Assertion

Ref	Expression
upcic	⊢ (𝜑 → 𝑋( ≃𝑐 ‘𝐷)𝑌)

Distinct variable groups:   𝑣,𝐵   𝑤,𝐵   𝑓,𝐹,𝑘,𝑤   𝑔,𝐹,𝑙,𝑣   𝑓,𝐺,𝑘,𝑤   𝑔,𝐺,𝑙,𝑣   𝑓,𝐻,𝑘,𝑤   𝑔,𝐻,𝑙,𝑣   𝑓,𝐽,𝑤   𝑔,𝐽,𝑣   𝑓,𝑀,𝑘,𝑤   𝑔,𝑀,𝑙   𝑓,𝑁,𝑘   𝑔,𝑁,𝑙,𝑣   𝑓,𝑂,𝑘,𝑤   𝑔,𝑂,𝑙,𝑣   𝑓,𝑋,𝑘,𝑤   𝑔,𝑋,𝑙,𝑣   𝑓,𝑌,𝑘,𝑤   𝑔,𝑌,𝑙,𝑣   𝑓,𝑍,𝑘,𝑤   𝑔,𝑍,𝑙,𝑣

Allowed substitution hints:   𝜑(𝑤,𝑣,𝑓,𝑔,𝑘,𝑙)   𝐵(𝑓,𝑔,𝑘,𝑙)   𝐶(𝑤,𝑣,𝑓,𝑔,𝑘,𝑙)   𝐷(𝑤,𝑣,𝑓,𝑔,𝑘,𝑙)   𝐸(𝑤,𝑣,𝑓,𝑔,𝑘,𝑙)   𝐽(𝑘,𝑙)   𝑀(𝑣)   𝑁(𝑤)

Proof of Theorem upcic

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		upcic.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
2		upcic.c	. . 3 ⊢ 𝐶 = (Base‘𝐸)
3		upcic.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐷)
4		upcic.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐸)
5		upcic.o	. . 3 ⊢ 𝑂 = (comp‘𝐸)
6		upcic.f	. . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
7		upcic.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		upcic.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9		upcic.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐶)
10		upcic.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ (𝑍𝐽(𝐹‘𝑋)))
11		upcic.1	. . 3 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ∀𝑓 ∈ (𝑍𝐽(𝐹‘𝑤))∃!𝑘 ∈ (𝑋𝐻𝑤)𝑓 = (((𝑋𝐺𝑤)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑤))𝑀))
12		upcic.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ (𝑍𝐽(𝐹‘𝑌)))
13		upcic.2	. . 3 ⊢ (𝜑 → ∀𝑣 ∈ 𝐵 ∀𝑔 ∈ (𝑍𝐽(𝐹‘𝑣))∃!𝑙 ∈ (𝑌𝐻𝑣)𝑔 = (((𝑌𝐺𝑣)‘𝑙)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑣))𝑁))
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	upciclem4 49827	. 2 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐷)𝑌 ∧ ∃𝑟 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)))
15	14	simpld 499	1 ⊢ (𝜑 → 𝑋( ≃𝑐 ‘𝐷)𝑌)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  ∃!wreu 3374  ⟨cop 4597   class class class wbr 5110  ‘cfv 6534  (class class class)co 7408  Basecbs 17265  Hom chom 17317  compcco 17318  Isociso 17799   ≃_𝑐 ccic 17848   Func cfunc 17907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7982  df-2nd 7983  df-supp 8153  df-map 8822  df-ixp 8892  df-cat 17720  df-cid 17721  df-sect 17800  df-inv 17801  df-iso 17802  df-cic 17849  df-func 17911

This theorem is referenced by: (None)