Theorem thincmod 49423

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 2831	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
5		thincmo.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6	5, 3	eleqtrd 2831	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
7		eqid 2730	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
8		eqid 2730	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
9	1, 4, 6, 7, 8	thincmo 49421	. 2 ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌))
10		thincn0eu.h	. . . . 5 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
11	10	oveqd 7407	. . . 4 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
12	11	eleq2d 2815	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↔ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)))
13	12	mobidv 2543	. 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 1540   ∈ wcel 2109  ∃*wmo 2532  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  Hom chom 17238  ThinCatcthinc 49410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-nul 5264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-ov 7393  df-thinc 49411

This theorem is referenced by:  thincn0eu  49424