Theorem upcic 49202

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 49201	. 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 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  ∃!wreu 3344  ⟨cop 4577   class class class wbr 5086  ‘cfv 6476  (class class class)co 7341  Basecbs 17115  Hom chom 17167  compcco 17168  Isociso 17648   ≃_𝑐 ccic 17697   Func cfunc 17756

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-supp 8086  df-map 8747  df-ixp 8817  df-cat 17569  df-cid 17570  df-sect 17649  df-inv 17650  df-iso 17651  df-cic 17698  df-func 17760

This theorem is referenced by: (None)