Theorem f1cdmsn 7237

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 6736	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→{𝐵} → 𝐹:𝐴⟶{𝐵})
2		fvconst 7117	. . . . . . . . 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 7117	. . . . . . . . 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 2774	. . . . . . 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 7211	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
9	8	3impb 1115	. . . . . 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 3128	. . 3 ⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ 𝑦 ∈ 𝐴) → ∀𝑧 ∈ 𝐴 𝑦 = 𝑧)
13	12	reximdva0 4295	. 2 ⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ 𝐴 ≠ ∅) → ∃𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 𝑦 = 𝑧)
14		issn 4775	. 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 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  ∃wrex 3061  ∅c0 4273  {csn 4567  ⟶wf 6494  –1-1→wf1 6495  ‘cfv 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fv 6506

This theorem is referenced by:  snnen2o  9155  sdom1  9160  1sdom2dom  9164