Theorem upeu 48901

Description: A universal property defines an essentially unique (strong form) pair of object 𝑋 and morphism 𝑀 if it exists. (Contributed by Zhi Wang, 19-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
upeu	⊢ (𝜑 → ∃!𝑟 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))

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

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

Proof of Theorem upeu

Step	Hyp	Ref	Expression
1		upcic.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
2		upcic.c	. . . 4 ⊢ 𝐶 = (Base‘𝐸)
3		upcic.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐷)
4		upcic.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐸)
5		upcic.o	. . . 4 ⊢ 𝑂 = (comp‘𝐸)
6		upcic.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
7		upcic.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		upcic.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9		upcic.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐶)
10		upcic.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑍𝐽(𝐹‘𝑋)))
11		upcic.1	. . . 4 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ∀𝑓 ∈ (𝑍𝐽(𝐹‘𝑤))∃!𝑘 ∈ (𝑋𝐻𝑤)𝑓 = (((𝑋𝐺𝑤)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑤))𝑀))
12		upcic.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (𝑍𝐽(𝐹‘𝑌)))
13		upcic.2	. . . 4 ⊢ (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 48899	. . 3 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐷)𝑌 ∧ ∃𝑟 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)))
15	14	simprd 495	. 2 ⊢ (𝜑 → ∃𝑟 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
16		eqid 2736	. . . 4 ⊢ (Iso‘𝐷) = (Iso‘𝐷)
17	6	funcrcl2 48885	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
18	1, 3, 16, 17, 7, 8	isohom 17816	. . 3 ⊢ (𝜑 → (𝑋(Iso‘𝐷)𝑌) ⊆ (𝑋𝐻𝑌))
19	11, 8, 12	upciclem1 48896	. . . 4 ⊢ (𝜑 → ∃!𝑟 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
20		reurmo 3382	. . . 4 ⊢ (∃!𝑟 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) → ∃*𝑟 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
21	19, 20	syl 17	. . 3 ⊢ (𝜑 → ∃*𝑟 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
22		nfcv 2904	. . . 4 ⊢ Ⅎ𝑟(𝑋(Iso‘𝐷)𝑌)
23		nfcv 2904	. . . 4 ⊢ Ⅎ𝑟(𝑋𝐻𝑌)
24	22, 23	ssrmof 4050	. . 3 ⊢ ((𝑋(Iso‘𝐷)𝑌) ⊆ (𝑋𝐻𝑌) → (∃*𝑟 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) → ∃*𝑟 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)))
25	18, 21, 24	sylc 65	. 2 ⊢ (𝜑 → ∃*𝑟 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
26		reu5 3381	. 2 ⊢ (∃!𝑟 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) ↔ (∃𝑟 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) ∧ ∃*𝑟 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)))
27	15, 25, 26	sylanbrc 583	1 ⊢ (𝜑 → ∃!𝑟 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ∀wral 3060  ∃wrex 3069  ∃!wreu 3377  ∃*wrmo 3378   ⊆ wss 3950  ⟨cop 4630   class class class wbr 5141  ‘cfv 6559  (class class class)co 7429  Basecbs 17243  Hom chom 17304  compcco 17305  Isociso 17786   ≃_𝑐 ccic 17835   Func cfunc 17895

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-riota 7386  df-ov 7432  df-oprab 7433  df-mpo 7434  df-1st 8010  df-2nd 8011  df-supp 8182  df-map 8864  df-ixp 8934  df-cat 17707  df-cid 17708  df-sect 17787  df-inv 17788  df-iso 17789  df-cic 17836  df-func 17899

This theorem is referenced by:  upeu3  48919