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Theorem termoeu1w 16875
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 16874 . . 3 (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
5 euex 2641 . . 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 2770 . . 3 (Iso‘𝐶) = (Iso‘𝐶)
8 eqid 2770 . . 3 (Base‘𝐶) = (Base‘𝐶)
9 termoo 16864 . . . 4 (𝐶 ∈ Cat → (𝐴 ∈ (TermO‘𝐶) → 𝐴 ∈ (Base‘𝐶)))
101, 2, 9sylc 65 . . 3 (𝜑𝐴 ∈ (Base‘𝐶))
11 termoo 16864 . . . 4 (𝐶 ∈ Cat → (𝐵 ∈ (TermO‘𝐶) → 𝐵 ∈ (Base‘𝐶)))
121, 3, 11sylc 65 . . 3 (𝜑𝐵 ∈ (Base‘𝐶))
137, 8, 1, 10, 12cic 16665 . 2 (𝜑 → (𝐴( ≃𝑐𝐶)𝐵 ↔ ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
146, 13mpbird 247 1 (𝜑𝐴( ≃𝑐𝐶)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1851  wcel 2144  ∃!weu 2617   class class class wbr 4784  cfv 6031  (class class class)co 6792  Basecbs 16063  Catccat 16531  Isociso 16612  𝑐 ccic 16661  TermOctermo 16845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-supp 7446  df-cat 16535  df-cid 16536  df-sect 16613  df-inv 16614  df-iso 16615  df-cic 16662  df-termo 16848
This theorem is referenced by:  nzerooringczr  42590
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