Theorem f1cdmsn 7280

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 6788	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→{𝐵} → 𝐹:𝐴⟶{𝐵})
2		fvconst 7162	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = 𝐵)
3	2	3adant3 1133	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑦) = 𝐵)
4		fvconst 7162	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = 𝐵)
5	4	3adant2 1132	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = 𝐵)
6	3, 5	eqtr4d 2776	. . . . . . 7 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑧))
7	1, 6	syl3an1 1164	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑧))
8		f1veqaeq 7256	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
9	8	3impb 1116	. . . . . 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 1122	. . . 4 ⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ 𝑦 ∈ 𝐴) → (𝑧 ∈ 𝐴 → 𝑦 = 𝑧))
12	11	ralrimiv 3146	. . 3 ⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ 𝑦 ∈ 𝐴) → ∀𝑧 ∈ 𝐴 𝑦 = 𝑧)
13	12	reximdva0 4352	. 2 ⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ 𝐴 ≠ ∅) → ∃𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 𝑦 = 𝑧)
14		issn 4834	. 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 397   ∧ w3a 1088   = wceq 1542  ∃wex 1782   ∈ wcel 2107   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  ∅c0 4323  {csn 4629  ⟶wf 6540  –1-1→wf1 6541  ‘cfv 6544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fv 6552

This theorem is referenced by:  snnen2o  9237  sdom1  9242  1sdom2dom  9247