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Theorem termoeu1w 17274
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 17273 . . 3 (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
5 euex 2660 . . 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 2826 . . 3 (Iso‘𝐶) = (Iso‘𝐶)
8 eqid 2826 . . 3 (Base‘𝐶) = (Base‘𝐶)
9 termoo 17263 . . . 4 (𝐶 ∈ Cat → (𝐴 ∈ (TermO‘𝐶) → 𝐴 ∈ (Base‘𝐶)))
101, 2, 9sylc 65 . . 3 (𝜑𝐴 ∈ (Base‘𝐶))
11 termoo 17263 . . . 4 (𝐶 ∈ Cat → (𝐵 ∈ (TermO‘𝐶) → 𝐵 ∈ (Base‘𝐶)))
121, 3, 11sylc 65 . . 3 (𝜑𝐵 ∈ (Base‘𝐶))
137, 8, 1, 10, 12cic 17064 . 2 (𝜑 → (𝐴( ≃𝑐𝐶)𝐵 ↔ ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
146, 13mpbird 258 1 (𝜑𝐴( ≃𝑐𝐶)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1773  wcel 2107  ∃!weu 2651   class class class wbr 5063  cfv 6354  (class class class)co 7150  Basecbs 16478  Catccat 16930  Isociso 17011  𝑐 ccic 17060  TermOctermo 17244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7685  df-2nd 7686  df-supp 7827  df-cat 16934  df-cid 16935  df-sect 17012  df-inv 17013  df-iso 17014  df-cic 17061  df-termo 17247
This theorem is referenced by:  nzerooringczr  44245
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