Theorem upeu 49646

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 49644	. . 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 49554	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
18	1, 3, 16, 17, 7, 8	isohom 17743	. . 3 ⊢ (𝜑 → (𝑋(Iso‘𝐷)𝑌) ⊆ (𝑋𝐻𝑌))
19	11, 8, 12	upciclem1 49641	. . . 4 ⊢ (𝜑 → ∃!𝑟 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
20		reurmo 3345	. . . 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 2898	. . . 4 ⊢ Ⅎ𝑟(𝑋(Iso‘𝐷)𝑌)
23		nfcv 2898	. . . 4 ⊢ Ⅎ𝑟(𝑋𝐻𝑌)
24	22, 23	ssrmof 3989	. . 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 3344	. 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 584	1 ⊢ (𝜑 → ∃!𝑟 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  ∃!wreu 3340  ∃*wrmo 3341   ⊆ wss 3889  ⟨cop 4573   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  Hom chom 17231  compcco 17232  Isociso 17713   ≃_𝑐 ccic 17762   Func cfunc 17821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-supp 8111  df-map 8775  df-ixp 8846  df-cat 17634  df-cid 17635  df-sect 17714  df-inv 17715  df-iso 17716  df-cic 17763  df-func 17825

This theorem is referenced by:  upeu3  49670