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Theorem thincmo2 48217
Description: Morphisms in the same hom-set are identical. (Contributed by Zhi Wang, 17-Sep-2024.)
Hypotheses
Ref Expression
isthincd2lem1.1 (𝜑𝑋𝐵)
isthincd2lem1.2 (𝜑𝑌𝐵)
isthincd2lem1.3 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
isthincd2lem1.4 (𝜑𝐺 ∈ (𝑋𝐻𝑌))
thincmo2.b 𝐵 = (Base‘𝐶)
thincmo2.h 𝐻 = (Hom ‘𝐶)
thincmo2.c (𝜑𝐶 ∈ ThinCat)
Assertion
Ref Expression
thincmo2 (𝜑𝐹 = 𝐺)

Proof of Theorem thincmo2
Dummy variables 𝑦 𝑥 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isthincd2lem1.1 . 2 (𝜑𝑋𝐵)
2 isthincd2lem1.2 . 2 (𝜑𝑌𝐵)
3 isthincd2lem1.3 . 2 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
4 isthincd2lem1.4 . 2 (𝜑𝐺 ∈ (𝑋𝐻𝑌))
5 thincmo2.c . . 3 (𝜑𝐶 ∈ ThinCat)
6 thincmo2.b . . . . 5 𝐵 = (Base‘𝐶)
7 thincmo2.h . . . . 5 𝐻 = (Hom ‘𝐶)
86, 7isthinc 48210 . . . 4 (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
98simprbi 495 . . 3 (𝐶 ∈ ThinCat → ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
105, 9syl 17 . 2 (𝜑 → ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
111, 2, 3, 4, 10isthincd2lem1 48216 1 (𝜑𝐹 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  ∃*wmo 2526  wral 3050  cfv 6549  (class class class)co 7419  Basecbs 17183  Hom chom 17247  Catccat 17647  ThinCatcthinc 48208
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-ov 7422  df-thinc 48209
This theorem is referenced by:  thincmo  48218  thincid  48222  thincmon  48223  thincepi  48224  functhinclem4  48233
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