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Theorem thincmo2 46027
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 46020 . . . 4 (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
98simprbi 500 . . 3 (𝐶 ∈ ThinCat → ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
105, 9syl 17 . 2 (𝜑 → ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
111, 2, 3, 4, 10isthincd2lem1 46026 1 (𝜑𝐹 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2112  ∃*wmo 2539  wral 3064  cfv 6400  (class class class)co 7234  Basecbs 16792  Hom chom 16845  Catccat 17199  ThinCatcthinc 46018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-nul 5215
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3425  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4456  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4836  df-br 5070  df-iota 6358  df-fv 6408  df-ov 7237  df-thinc 46019
This theorem is referenced by:  thincmo  46028  thincid  46032  thincmon  46033  thincepi  46034  functhinclem4  46043
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