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Theorem termoeu1w 18073
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 18072 . . 3 (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
5 euex 2575 . . 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 2735 . . 3 (Iso‘𝐶) = (Iso‘𝐶)
8 eqid 2735 . . 3 (Base‘𝐶) = (Base‘𝐶)
9 termoo 18062 . . . 4 (𝐶 ∈ Cat → (𝐴 ∈ (TermO‘𝐶) → 𝐴 ∈ (Base‘𝐶)))
101, 2, 9sylc 65 . . 3 (𝜑𝐴 ∈ (Base‘𝐶))
11 termoo 18062 . . . 4 (𝐶 ∈ Cat → (𝐵 ∈ (TermO‘𝐶) → 𝐵 ∈ (Base‘𝐶)))
121, 3, 11sylc 65 . . 3 (𝜑𝐵 ∈ (Base‘𝐶))
137, 8, 1, 10, 12cic 17847 . 2 (𝜑 → (𝐴( ≃𝑐𝐶)𝐵 ↔ ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
146, 13mpbird 257 1 (𝜑𝐴( ≃𝑐𝐶)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1776  wcel 2106  ∃!weu 2566   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  Catccat 17709  Isociso 17794  𝑐 ccic 17843  TermOctermo 18036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-supp 8185  df-cat 17713  df-cid 17714  df-sect 17795  df-inv 17796  df-iso 17797  df-cic 17844  df-termo 18039
This theorem is referenced by:  nzerooringczr  21509
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