Theorem thincmod 49462

Description: At most one morphism in each hom-set (deduction form). (Contributed by Zhi Wang, 21-Sep-2024.)

Hypotheses

Ref	Expression
thincmo.c	⊢ (𝜑 → 𝐶 ∈ ThinCat)
thincmo.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
thincmo.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
thincn0eu.b	⊢ (𝜑 → 𝐵 = (Base‘𝐶))
thincn0eu.h	⊢ (𝜑 → 𝐻 = (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

Step	Hyp	Ref	Expression
1		thincmo.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
2		thincmo.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3		thincn0eu.b	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
4	2, 3	eleqtrd 2833	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
5		thincmo.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6	5, 3	eleqtrd 2833	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
7		eqid 2731	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
8		eqid 2731	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
9	1, 4, 6, 7, 8	thincmo 49460	. 2 ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌))
10		thincn0eu.h	. . . . 5 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
11	10	oveqd 7358	. . . 4 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
12	11	eleq2d 2817	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↔ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)))
13	12	mobidv 2544	. 2 ⊢ (𝜑 → (∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌) ↔ ∃*𝑓 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)))
14	9, 13	mpbird 257	1 ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ∃*wmo 2533  ‘cfv 6476  (class class class)co 7341  Basecbs 17115  Hom chom 17167  ThinCatcthinc 49449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-nul 5239

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-iota 6432  df-fv 6484  df-ov 7344  df-thinc 49450

This theorem is referenced by:  thincn0eu  49463