Theorem f1cdmsn 7302

Description: If a one-to-one function with a nonempty domain has a singleton as its codomain, its domain must also be a singleton. (Contributed by BTernaryTau, 1-Dec-2024.)

Assertion

Ref	Expression
f1cdmsn	⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ 𝐴 ≠ ∅) → ∃𝑥 𝐴 = {𝑥})

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem f1cdmsn

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1f 6805	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→{𝐵} → 𝐹:𝐴⟶{𝐵})
2		fvconst 7184	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = 𝐵)
3	2	3adant3 1131	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑦) = 𝐵)
4		fvconst 7184	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = 𝐵)
5	4	3adant2 1130	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = 𝐵)
6	3, 5	eqtr4d 2778	. . . . . . 7 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑧))
7	1, 6	syl3an1 1162	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑧))
8		f1veqaeq 7277	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
9	8	3impb 1114	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
10	7, 9	mpd 15	. . . . 5 ⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → 𝑦 = 𝑧)
11	10	3expia 1120	. . . 4 ⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ 𝑦 ∈ 𝐴) → (𝑧 ∈ 𝐴 → 𝑦 = 𝑧))
12	11	ralrimiv 3143	. . 3 ⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ 𝑦 ∈ 𝐴) → ∀𝑧 ∈ 𝐴 𝑦 = 𝑧)
13	12	reximdva0 4361	. 2 ⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ 𝐴 ≠ ∅) → ∃𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 𝑦 = 𝑧)
14		issn 4837	. 2 ⊢ (∃𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 𝑦 = 𝑧 → ∃𝑥 𝐴 = {𝑥})
15	13, 14	syl 17	1 ⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ 𝐴 ≠ ∅) → ∃𝑥 𝐴 = {𝑥})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537  ∃wex 1776   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∃wrex 3068  ∅c0 4339  {csn 4631  ⟶wf 6559  –1-1→wf1 6560  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fv 6571

This theorem is referenced by:  snnen2o  9271  sdom1  9276  1sdom2dom  9281