Theorem termoeu1w 17974

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 17973	. . 3 ⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
5		euex 2570	. . 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 2731	. . 3 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
8		eqid 2731	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		termoo 17963	. . . 4 ⊢ (𝐶 ∈ Cat → (𝐴 ∈ (TermO‘𝐶) → 𝐴 ∈ (Base‘𝐶)))
10	1, 2, 9	sylc 65	. . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐶))
11		termoo 17963	. . . 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 17751	. 2 ⊢ (𝜑 → (𝐴( ≃𝑐 ‘𝐶)𝐵 ↔ ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
14	6, 13	mpbird 256	1 ⊢ (𝜑 → 𝐴( ≃𝑐 ‘𝐶)𝐵)

Syntax hints:   → wi 4  ∃wex 1780   ∈ wcel 2105  ∃!weu 2561   class class class wbr 5149  ‘cfv 6544  (class class class)co 7412  Basecbs 17149  Catccat 17613  Isociso 17698   ≃_𝑐 ccic 17747  TermOctermo 17937

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7978  df-2nd 7979  df-supp 8150  df-cat 17617  df-cid 17618  df-sect 17699  df-inv 17700  df-iso 17701  df-cic 17748  df-termo 17940

This theorem is referenced by:  nzerooringczr  47060