Users' Mathboxes Mathbox for Zhi Wang < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eufsn Structured version   Visualization version   GIF version

Theorem eufsn 45683
Description: There is exactly one function into a singleton, assuming ax-rep 5151. See eufsn2 45684 for different axiom requirements. If existence is not needed, use mofsn 45685 or mofsn2 45686 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024.)
Hypotheses
Ref Expression
eufsn.1 (𝜑𝐵𝑊)
eufsn.2 (𝜑𝐴𝑉)
Assertion
Ref Expression
eufsn (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝜑,𝑓
Allowed substitution hints:   𝑉(𝑓)   𝑊(𝑓)

Proof of Theorem eufsn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eufsn.1 . 2 (𝜑𝐵𝑊)
2 eufsn.2 . . 3 (𝜑𝐴𝑉)
3 fconstmpt 5579 . . . 4 (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
4 mptexg 6988 . . . 4 (𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)
53, 4eqeltrid 2837 . . 3 (𝐴𝑉 → (𝐴 × {𝐵}) ∈ V)
62, 5syl 17 . 2 (𝜑 → (𝐴 × {𝐵}) ∈ V)
71, 6eufsnlem 45682 1 (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2113  ∃!weu 2569  Vcvv 3397  {csn 4513  cmpt 5107   × cxp 5517  wf 6329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pr 5293
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator