Theorem upcic 48926

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 48925	. 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 1539   ∈ wcel 2107  ∀wral 3050  ∃wrex 3059  ∃!wreu 3355  ⟨cop 4605   class class class wbr 5116  ‘cfv 6527  (class class class)co 7399  Basecbs 17213  Hom chom 17267  compcco 17268  Isociso 17744   ≃_𝑐 ccic 17793   Func cfunc 17852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5246  ax-sep 5263  ax-nul 5273  ax-pow 5332  ax-pr 5399  ax-un 7723

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3357  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-iun 4966  df-br 5117  df-opab 5179  df-mpt 5199  df-id 5545  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6480  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7356  df-ov 7402  df-oprab 7403  df-mpo 7404  df-1st 7982  df-2nd 7983  df-supp 8154  df-map 8836  df-ixp 8906  df-cat 17665  df-cid 17666  df-sect 17745  df-inv 17746  df-iso 17747  df-cic 17794  df-func 17856

This theorem is referenced by: (None)