Theorem upcic 49155

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 49154	. 2 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐷)𝑌 ∧ ∃𝑟 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)))
15	14	simpld 494	1 ⊢ (𝜑 → 𝑋( ≃𝑐 ‘𝐷)𝑌)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  ∃!wreu 3341  ⟨cop 4583   class class class wbr 5092  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  Hom chom 17172  compcco 17173  Isociso 17653   ≃_𝑐 ccic 17702   Func cfunc 17761

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-supp 8094  df-map 8755  df-ixp 8825  df-cat 17574  df-cid 17575  df-sect 17654  df-inv 17655  df-iso 17656  df-cic 17703  df-func 17765

This theorem is referenced by: (None)