Theorem termoeu1w 18098

Description: Terminal objects are essentially unique (weak form), i.e. if A and B are terminal objects, then A and B are isomorphic. Proposition 7.6 of [Adamek] p. 103. (Contributed by AV, 18-Apr-2020.)

Hypotheses

Ref	Expression
termoeu1.c	⊢ (𝜑 → 𝐶 ∈ Cat)
termoeu1.a	⊢ (𝜑 → 𝐴 ∈ (TermO‘𝐶))
termoeu1.b	⊢ (𝜑 → 𝐵 ∈ (TermO‘𝐶))

Assertion

Ref	Expression
termoeu1w	⊢ (𝜑 → 𝐴( ≃𝑐 ‘𝐶)𝐵)

Proof of Theorem termoeu1w

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		termoeu1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
2		termoeu1.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (TermO‘𝐶))
3		termoeu1.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (TermO‘𝐶))
4	1, 2, 3	termoeu1 18097	. . 3 ⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
5		euex 2607	. . 3 ⊢ (∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵) → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
6	4, 5	syl 18	. 2 ⊢ (𝜑 → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
7		eqid 2765	. . 3 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
8		eqid 2765	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		termoo 18087	. . . 4 ⊢ (𝐶 ∈ Cat → (𝐴 ∈ (TermO‘𝐶) → 𝐴 ∈ (Base‘𝐶)))
10	1, 2, 9	sylc 66	. . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐶))
11		termoo 18087	. . . 4 ⊢ (𝐶 ∈ Cat → (𝐵 ∈ (TermO‘𝐶) → 𝐵 ∈ (Base‘𝐶)))
12	1, 3, 11	sylc 66	. . 3 ⊢ (𝜑 → 𝐵 ∈ (Base‘𝐶))
13	7, 8, 1, 10, 12	cic 17878	. 2 ⊢ (𝜑 → (𝐴( ≃𝑐 ‘𝐶)𝐵 ↔ ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
14	6, 13	mpbird 260	1 ⊢ (𝜑 → 𝐴( ≃𝑐 ‘𝐶)𝐵)

This proof depends on syntax axioms:   → wi 4  ∃wex 1812   ∈ wcel 2146  ∃!weu 2598   class class class wbr 5111  ‘cfv 6540  (class class class)co 7419  Basecbs 17291  Catccat 17742  Isociso 17825   ≃_𝑐 ccic 17874  TermOctermo 18061

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-supp 8163  df-cat 17746  df-cid 17747  df-sect 17826  df-inv 17827  df-iso 17828  df-cic 17875  df-termo 18064

This theorem is used by:  nzerooringczr  21680  termcterm2  50349  termcciso  50351