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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:𝐴1-1→{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.
StepHypRef Expression
1 f1f 6804 . . . . . . 7 (𝐹:𝐴1-1→{𝐵} → 𝐹:𝐴⟶{𝐵})
2 fvconst 7184 . . . . . . . . 9 ((𝐹:𝐴⟶{𝐵} ∧ 𝑦𝐴) → (𝐹𝑦) = 𝐵)
323adant3 1133 . . . . . . . 8 ((𝐹:𝐴⟶{𝐵} ∧ 𝑦𝐴𝑧𝐴) → (𝐹𝑦) = 𝐵)
4 fvconst 7184 . . . . . . . . 9 ((𝐹:𝐴⟶{𝐵} ∧ 𝑧𝐴) → (𝐹𝑧) = 𝐵)
543adant2 1132 . . . . . . . 8 ((𝐹:𝐴⟶{𝐵} ∧ 𝑦𝐴𝑧𝐴) → (𝐹𝑧) = 𝐵)
63, 5eqtr4d 2780 . . . . . . 7 ((𝐹:𝐴⟶{𝐵} ∧ 𝑦𝐴𝑧𝐴) → (𝐹𝑦) = (𝐹𝑧))
71, 6syl3an1 1164 . . . . . 6 ((𝐹:𝐴1-1→{𝐵} ∧ 𝑦𝐴𝑧𝐴) → (𝐹𝑦) = (𝐹𝑧))
8 f1veqaeq 7277 . . . . . . 7 ((𝐹:𝐴1-1→{𝐵} ∧ (𝑦𝐴𝑧𝐴)) → ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
983impb 1115 . . . . . 6 ((𝐹:𝐴1-1→{𝐵} ∧ 𝑦𝐴𝑧𝐴) → ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
107, 9mpd 15 . . . . 5 ((𝐹:𝐴1-1→{𝐵} ∧ 𝑦𝐴𝑧𝐴) → 𝑦 = 𝑧)
11103expia 1122 . . . 4 ((𝐹:𝐴1-1→{𝐵} ∧ 𝑦𝐴) → (𝑧𝐴𝑦 = 𝑧))
1211ralrimiv 3145 . . 3 ((𝐹:𝐴1-1→{𝐵} ∧ 𝑦𝐴) → ∀𝑧𝐴 𝑦 = 𝑧)
1312reximdva0 4355 . 2 ((𝐹:𝐴1-1→{𝐵} ∧ 𝐴 ≠ ∅) → ∃𝑦𝐴𝑧𝐴 𝑦 = 𝑧)
14 issn 4832 . 2 (∃𝑦𝐴𝑧𝐴 𝑦 = 𝑧 → ∃𝑥 𝐴 = {𝑥})
1513, 14syl 17 1 ((𝐹:𝐴1-1→{𝐵} ∧ 𝐴 ≠ ∅) → ∃𝑥 𝐴 = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  wrex 3070  c0 4333  {csn 4626  wf 6557  1-1wf1 6558  cfv 6561
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569
This theorem is referenced by:  snnen2o  9273  sdom1  9278  1sdom2dom  9283
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