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Theorem termoeu1w 16934
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 (𝜑𝐶 ∈ Cat)
termoeu1.a (𝜑𝐴 ∈ (TermO‘𝐶))
termoeu1.b (𝜑𝐵 ∈ (TermO‘𝐶))
Assertion
Ref Expression
termoeu1w (𝜑𝐴( ≃𝑐𝐶)𝐵)

Proof of Theorem termoeu1w
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 termoeu1.c . . . 4 (𝜑𝐶 ∈ Cat)
2 termoeu1.a . . . 4 (𝜑𝐴 ∈ (TermO‘𝐶))
3 termoeu1.b . . . 4 (𝜑𝐵 ∈ (TermO‘𝐶))
41, 2, 3termoeu1 16933 . . 3 (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
5 euex 2591 . . 3 (∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵) → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
64, 5syl 17 . 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 16923 . . . 4 (𝐶 ∈ Cat → (𝐴 ∈ (TermO‘𝐶) → 𝐴 ∈ (Base‘𝐶)))
101, 2, 9sylc 65 . . 3 (𝜑𝐴 ∈ (Base‘𝐶))
11 termoo 16923 . . . 4 (𝐶 ∈ Cat → (𝐵 ∈ (TermO‘𝐶) → 𝐵 ∈ (Base‘𝐶)))
121, 3, 11sylc 65 . . 3 (𝜑𝐵 ∈ (Base‘𝐶))
137, 8, 1, 10, 12cic 16724 . 2 (𝜑 → (𝐴( ≃𝑐𝐶)𝐵 ↔ ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
146, 13mpbird 248 1 (𝜑𝐴( ≃𝑐𝐶)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1874  wcel 2155  ∃!weu 2581   class class class wbr 4809  cfv 6068  (class class class)co 6842  Basecbs 16130  Catccat 16590  Isociso 16671  𝑐 ccic 16720  TermOctermo 16904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-supp 7498  df-cat 16594  df-cid 16595  df-sect 16672  df-inv 16673  df-iso 16674  df-cic 16721  df-termo 16907
This theorem is referenced by:  nzerooringczr  42741
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