Theorem thincmo 49932

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 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑋 ∈ 𝐵)
3		thincmo.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
4	3	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑌 ∈ 𝐵)
5		simprl 777	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑓 ∈ (𝑋𝐻𝑌))
6		simprr 779	. . . . 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 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝐶 ∈ ThinCat)
11	2, 4, 5, 6, 7, 8, 10	thincmo2 49930	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑓 = 𝑔)
12	11	ex 414	. . 3 ⊢ (𝜑 → ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝑔))
13	12	alrimivv 1936	. 2 ⊢ (𝜑 → ∀𝑓∀𝑔((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝑔))
14		eleq1w 2824	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ (𝑋𝐻𝑌) ↔ 𝑔 ∈ (𝑋𝐻𝑌)))
15	14	mo4 2572	. 2 ⊢ (∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌) ↔ ∀𝑓∀𝑔((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝑔))
16	13, 15	sylibr 236	1 ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))

Syntax hints:   → wi 4   ∧ wa 397  ∀wal 1546   = wceq 1548   ∈ wcel 2121  ∃*wmo 2543  ‘cfv 6489  (class class class)co 7360  Basecbs 17174  Hom chom 17226  ThinCatcthinc 49921

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 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-nul 5231

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-iota 6445  df-fv 6497  df-ov 7363  df-thinc 49922

This theorem is referenced by:  thincmod  49934  oppcthin  49942  subthinc  49947  functhinclem1  49948  functhinclem4  49951  thincfth  49956  thincciso2  49959