Theorem f1cdmsn 7275

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 6780	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→{𝐵} → 𝐹:𝐴⟶{𝐵})
2		fvconst 7157	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = 𝐵)
3	2	3adant3 1129	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑦) = 𝐵)
4		fvconst 7157	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = 𝐵)
5	4	3adant2 1128	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = 𝐵)
6	3, 5	eqtr4d 2769	. . . . . . 7 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑧))
7	1, 6	syl3an1 1160	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑧))
8		f1veqaeq 7251	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
9	8	3impb 1112	. . . . . 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 1118	. . . 4 ⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ 𝑦 ∈ 𝐴) → (𝑧 ∈ 𝐴 → 𝑦 = 𝑧))
12	11	ralrimiv 3139	. . 3 ⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ 𝑦 ∈ 𝐴) → ∀𝑧 ∈ 𝐴 𝑦 = 𝑧)
13	12	reximdva0 4346	. 2 ⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ 𝐴 ≠ ∅) → ∃𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 𝑦 = 𝑧)
14		issn 4828	. 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 1084   = wceq 1533  ∃wex 1773   ∈ wcel 2098   ≠ wne 2934  ∀wral 3055  ∃wrex 3064  ∅c0 4317  {csn 4623  ⟶wf 6532  –1-1→wf1 6533  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fv 6544

This theorem is referenced by:  snnen2o  9236  sdom1  9241  1sdom2dom  9246