Theorem thincmod 49675

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 2838	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
5		thincmo.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6	5, 3	eleqtrd 2838	. . 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 49673	. 2 ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌))
10		thincn0eu.h	. . . . 5 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
11	10	oveqd 7375	. . . 4 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
12	11	eleq2d 2822	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↔ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)))
13	12	mobidv 2549	. 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 2113  ∃*wmo 2537  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  Hom chom 17188  ThinCatcthinc 49662

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-nul 5251

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 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361  df-thinc 49663

This theorem is referenced by:  thincn0eu  49676