Theorem upcic 49415

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 49414	. 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 2113  ∀wral 3051  ∃wrex 3060  ∃!wreu 3348  ⟨cop 4586   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  Hom chom 17188  compcco 17189  Isociso 17670   ≃_𝑐 ccic 17719   Func cfunc 17778

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-supp 8103  df-map 8765  df-ixp 8836  df-cat 17591  df-cid 17592  df-sect 17671  df-inv 17672  df-iso 17673  df-cic 17720  df-func 17782

This theorem is referenced by: (None)