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Theorem termoeu1w 17734
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 17733 . . 3 (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
5 euex 2577 . . 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 2738 . . 3 (Iso‘𝐶) = (Iso‘𝐶)
8 eqid 2738 . . 3 (Base‘𝐶) = (Base‘𝐶)
9 termoo 17723 . . . 4 (𝐶 ∈ Cat → (𝐴 ∈ (TermO‘𝐶) → 𝐴 ∈ (Base‘𝐶)))
101, 2, 9sylc 65 . . 3 (𝜑𝐴 ∈ (Base‘𝐶))
11 termoo 17723 . . . 4 (𝐶 ∈ Cat → (𝐵 ∈ (TermO‘𝐶) → 𝐵 ∈ (Base‘𝐶)))
121, 3, 11sylc 65 . . 3 (𝜑𝐵 ∈ (Base‘𝐶))
137, 8, 1, 10, 12cic 17511 . 2 (𝜑 → (𝐴( ≃𝑐𝐶)𝐵 ↔ ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
146, 13mpbird 256 1 (𝜑𝐴( ≃𝑐𝐶)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1782  wcel 2106  ∃!weu 2568   class class class wbr 5074  cfv 6433  (class class class)co 7275  Basecbs 16912  Catccat 17373  Isociso 17458  𝑐 ccic 17507  TermOctermo 17697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-supp 7978  df-cat 17377  df-cid 17378  df-sect 17459  df-inv 17460  df-iso 17461  df-cic 17508  df-termo 17700
This theorem is referenced by:  nzerooringczr  45630
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