Theorem thincmod 45928

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 2736	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
8		eqid 2736	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
9	1, 4, 6, 7, 8	thincmo 45926	. 2 ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌))
10		thincn0eu.h	. . . . 5 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
11	10	oveqd 7208	. . . 4 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
12	11	eleq2d 2816	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↔ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)))
13	12	mobidv 2548	. 2 ⊢ (𝜑 → (∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌) ↔ ∃*𝑓 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)))
14	9, 13	mpbird 260	1 ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2112  ∃*wmo 2537  ‘cfv 6358  (class class class)co 7191  Basecbs 16666  Hom chom 16760  ThinCatcthinc 45916

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 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-nul 5184

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 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-iota 6316  df-fv 6366  df-ov 7194  df-thinc 45917

This theorem is referenced by:  thincn0eu  45929