Theorem thincmo 50050

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 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑋 ∈ 𝐵)
3		thincmo.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
4	3	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑌 ∈ 𝐵)
5		simprl 780	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑓 ∈ (𝑋𝐻𝑌))
6		simprr 782	. . . . 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 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝐶 ∈ ThinCat)
11	2, 4, 5, 6, 7, 8, 10	thincmo2 50048	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑓 = 𝑔)
12	11	ex 416	. . 3 ⊢ (𝜑 → ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝑔))
13	12	alrimivv 1949	. 2 ⊢ (𝜑 → ∀𝑓∀𝑔((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝑔))
14		eleq1w 2846	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ (𝑋𝐻𝑌) ↔ 𝑔 ∈ (𝑋𝐻𝑌)))
15	14	mo4 2594	. 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 399  ∀wal 1559   = wceq 1561   ∈ wcel 2143  ∃*wmo 2565  ‘cfv 6522  (class class class)co 7397  Basecbs 17246  Hom chom 17298  ThinCatcthinc 50039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-nul 5257

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-iota 6478  df-fv 6530  df-ov 7400  df-thinc 50040

This theorem is referenced by:  thincmod  50052  oppcthin  50060  subthinc  50065  functhinclem1  50066  functhinclem4  50069  thincfth  50074  thincciso2  50077