Theorem thincmo2 49395

Description: Morphisms in the same hom-set are identical. (Contributed by Zhi Wang, 17-Sep-2024.)

Hypotheses

Ref	Expression
isthincd2lem1.1	⊢ (𝜑 → 𝑋 ∈ 𝐵)
isthincd2lem1.2	⊢ (𝜑 → 𝑌 ∈ 𝐵)
isthincd2lem1.3	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
isthincd2lem1.4	⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))
thincmo2.b	⊢ 𝐵 = (Base‘𝐶)
thincmo2.h	⊢ 𝐻 = (Hom ‘𝐶)
thincmo2.c	⊢ (𝜑 → 𝐶 ∈ ThinCat)

Assertion

Ref	Expression
thincmo2	⊢ (𝜑 → 𝐹 = 𝐺)

Proof of Theorem thincmo2

Dummy variables 𝑦 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isthincd2lem1.1	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
2		isthincd2lem1.2	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
3		isthincd2lem1.3	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
4		isthincd2lem1.4	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))
5		thincmo2.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
6		thincmo2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
7		thincmo2.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
8	6, 7	isthinc 49388	. . . 4 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
9	8	simprbi 496	. . 3 ⊢ (𝐶 ∈ ThinCat → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
10	5, 9	syl 17	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
11	1, 2, 3, 4, 10	isthincd2lem1 49394	1 ⊢ (𝜑 → 𝐹 = 𝐺)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∃*wmo 2532  ∀wral 3045  ‘cfv 6513  (class class class)co 7389  Basecbs 17185  Hom chom 17237  Catccat 17631  ThinCatcthinc 49386

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 5263

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 3756  df-csb 3865  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-iota 6466  df-fv 6521  df-ov 7392  df-thinc 49387

This theorem is referenced by:  thinchom  49396  thincmo  49397  thincid  49401  thincmon  49402  thincepi  49403  oppcthinco  49408  oppcthinendcALT  49410  functhinclem4  49416  termchommo  49454  funcsn  49510