Theorem thincmo 46198

Description: There is at most one morphism in each hom-set. (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)
thincmo.b	⊢ 𝐵 = (Base‘𝐶)
thincmo.h	⊢ 𝐻 = (Hom ‘𝐶)

Assertion

Ref	Expression
thincmo	⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))

Distinct variable groups:   𝐵,𝑓   𝐶,𝑓   𝑓,𝐻   𝑓,𝑋   𝑓,𝑌   𝜑,𝑓

Proof of Theorem thincmo

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		thincmo.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
2	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑋 ∈ 𝐵)
3		thincmo.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
4	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑌 ∈ 𝐵)
5		simprl 767	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑓 ∈ (𝑋𝐻𝑌))
6		simprr 769	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑔 ∈ (𝑋𝐻𝑌))
7		thincmo.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
8		thincmo.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
9		thincmo.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
10	9	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝐶 ∈ ThinCat)
11	2, 4, 5, 6, 7, 8, 10	thincmo2 46197	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑓 = 𝑔)
12	11	ex 412	. . 3 ⊢ (𝜑 → ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝑔))
13	12	alrimivv 1932	. 2 ⊢ (𝜑 → ∀𝑓∀𝑔((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝑔))
14		eleq1w 2821	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ (𝑋𝐻𝑌) ↔ 𝑔 ∈ (𝑋𝐻𝑌)))
15	14	mo4 2566	. 2 ⊢ (∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌) ↔ ∀𝑓∀𝑔((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝑔))
16	13, 15	sylibr 233	1 ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537   = wceq 1539   ∈ wcel 2108  ∃*wmo 2538  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Hom chom 16899  ThinCatcthinc 46188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258  df-thinc 46189

This theorem is referenced by:  thincmod  46200  oppcthin  46208  subthinc  46209  functhinclem1  46210  functhinclem4  46213  thincfth  46217