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Theorem thincmod 48831
Description: At most one morphism in each hom-set (deduction form). (Contributed by Zhi Wang, 21-Sep-2024.)
Hypotheses
Ref Expression
thincmo.c (𝜑𝐶 ∈ ThinCat)
thincmo.x (𝜑𝑋𝐵)
thincmo.y (𝜑𝑌𝐵)
thincn0eu.b (𝜑𝐵 = (Base‘𝐶))
thincn0eu.h (𝜑𝐻 = (Hom ‘𝐶))
Assertion
Ref Expression
thincmod (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
Distinct variable groups:   𝐵,𝑓   𝐶,𝑓   𝑓,𝐻   𝑓,𝑋   𝑓,𝑌   𝜑,𝑓

Proof of Theorem thincmod
StepHypRef Expression
1 thincmo.c . . 3 (𝜑𝐶 ∈ ThinCat)
2 thincmo.x . . . 4 (𝜑𝑋𝐵)
3 thincn0eu.b . . . 4 (𝜑𝐵 = (Base‘𝐶))
42, 3eleqtrd 2841 . . 3 (𝜑𝑋 ∈ (Base‘𝐶))
5 thincmo.y . . . 4 (𝜑𝑌𝐵)
65, 3eleqtrd 2841 . . 3 (𝜑𝑌 ∈ (Base‘𝐶))
7 eqid 2735 . . 3 (Base‘𝐶) = (Base‘𝐶)
8 eqid 2735 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
91, 4, 6, 7, 8thincmo 48829 . 2 (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌))
10 thincn0eu.h . . . . 5 (𝜑𝐻 = (Hom ‘𝐶))
1110oveqd 7448 . . . 4 (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
1211eleq2d 2825 . . 3 (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↔ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)))
1312mobidv 2547 . 2 (𝜑 → (∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌) ↔ ∃*𝑓 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)))
149, 13mpbird 257 1 (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  ∃*wmo 2536  cfv 6563  (class class class)co 7431  Basecbs 17245  Hom chom 17309  ThinCatcthinc 48819
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-nul 5312
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-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-thinc 48820
This theorem is referenced by:  thincn0eu  48832
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