Theorem termoeu1w 17271

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 17270	. . 3 ⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
5		euex 2637	. . 3 ⊢ (∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵) → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
6	4, 5	syl 17	. 2 ⊢ (𝜑 → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
7		eqid 2798	. . 3 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
8		eqid 2798	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		termoo 17260	. . . 4 ⊢ (𝐶 ∈ Cat → (𝐴 ∈ (TermO‘𝐶) → 𝐴 ∈ (Base‘𝐶)))
10	1, 2, 9	sylc 65	. . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐶))
11		termoo 17260	. . . 4 ⊢ (𝐶 ∈ Cat → (𝐵 ∈ (TermO‘𝐶) → 𝐵 ∈ (Base‘𝐶)))
12	1, 3, 11	sylc 65	. . 3 ⊢ (𝜑 → 𝐵 ∈ (Base‘𝐶))
13	7, 8, 1, 10, 12	cic 17061	. 2 ⊢ (𝜑 → (𝐴( ≃𝑐 ‘𝐶)𝐵 ↔ ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
14	6, 13	mpbird 260	1 ⊢ (𝜑 → 𝐴( ≃𝑐 ‘𝐶)𝐵)

Syntax hints:   → wi 4  ∃wex 1781   ∈ wcel 2111  ∃!weu 2628   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  Catccat 16927  Isociso 17008   ≃_𝑐 ccic 17057  TermOctermo 17241

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-supp 7814  df-cat 16931  df-cid 16932  df-sect 17009  df-inv 17010  df-iso 17011  df-cic 17058  df-termo 17244

This theorem is referenced by:  nzerooringczr  44696