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Theorem thincmo 46321
Description: There is at most one morphism in each hom-set. (Contributed by Zhi Wang, 21-Sep-2024.)
Hypotheses
Ref Expression
thincmo.c (𝜑𝐶 ∈ ThinCat)
thincmo.x (𝜑𝑋𝐵)
thincmo.y (𝜑𝑌𝐵)
thincmo.b 𝐵 = (Base‘𝐶)
thincmo.h 𝐻 = (Hom ‘𝐶)
Assertion
Ref Expression
thincmo (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
Distinct variable groups:   𝐵,𝑓   𝐶,𝑓   𝑓,𝐻   𝑓,𝑋   𝑓,𝑌   𝜑,𝑓

Proof of Theorem thincmo
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 thincmo.x . . . . . 6 (𝜑𝑋𝐵)
21adantr 481 . . . . 5 ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑋𝐵)
3 thincmo.y . . . . . 6 (𝜑𝑌𝐵)
43adantr 481 . . . . 5 ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑌𝐵)
5 simprl 768 . . . . 5 ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑓 ∈ (𝑋𝐻𝑌))
6 simprr 770 . . . . 5 ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑔 ∈ (𝑋𝐻𝑌))
7 thincmo.b . . . . 5 𝐵 = (Base‘𝐶)
8 thincmo.h . . . . 5 𝐻 = (Hom ‘𝐶)
9 thincmo.c . . . . . 6 (𝜑𝐶 ∈ ThinCat)
109adantr 481 . . . . 5 ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝐶 ∈ ThinCat)
112, 4, 5, 6, 7, 8, 10thincmo2 46320 . . . 4 ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑓 = 𝑔)
1211ex 413 . . 3 (𝜑 → ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝑔))
1312alrimivv 1932 . 2 (𝜑 → ∀𝑓𝑔((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝑔))
14 eleq1w 2822 . . 3 (𝑓 = 𝑔 → (𝑓 ∈ (𝑋𝐻𝑌) ↔ 𝑔 ∈ (𝑋𝐻𝑌)))
1514mo4 2567 . 2 (∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌) ↔ ∀𝑓𝑔((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝑔))
1613, 15sylibr 233 1 (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1537   = wceq 1539  wcel 2107  ∃*wmo 2539  cfv 6437  (class class class)co 7284  Basecbs 16921  Hom chom 16982  ThinCatcthinc 46311
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-nul 5231
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 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5076  df-iota 6395  df-fv 6445  df-ov 7287  df-thinc 46312
This theorem is referenced by:  thincmod  46323  oppcthin  46331  subthinc  46332  functhinclem1  46333  functhinclem4  46336  thincfth  46340
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