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Theorem rnmptsn 34498
Description: The range of a function mapping to singletons. (Contributed by ML, 15-Jul-2020.)
Assertion
Ref Expression
rnmptsn ran (𝑥𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
Distinct variable groups:   𝑢,𝐴   𝑥,𝑢
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem rnmptsn
StepHypRef Expression
1 rnopab 5819 . 2 ran {⟨𝑥, 𝑢⟩ ∣ (𝑥𝐴𝑢 = {𝑥})} = {𝑢 ∣ ∃𝑥(𝑥𝐴𝑢 = {𝑥})}
2 df-mpt 5138 . . 3 (𝑥𝐴 ↦ {𝑥}) = {⟨𝑥, 𝑢⟩ ∣ (𝑥𝐴𝑢 = {𝑥})}
32rneqi 5800 . 2 ran (𝑥𝐴 ↦ {𝑥}) = ran {⟨𝑥, 𝑢⟩ ∣ (𝑥𝐴𝑢 = {𝑥})}
4 df-rex 3141 . . 3 (∃𝑥𝐴 𝑢 = {𝑥} ↔ ∃𝑥(𝑥𝐴𝑢 = {𝑥}))
54abbii 2883 . 2 {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}} = {𝑢 ∣ ∃𝑥(𝑥𝐴𝑢 = {𝑥})}
61, 3, 53eqtr4i 2851 1 ran (𝑥𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wrex 3136  {csn 4557  {copab 5119  cmpt 5137  ran crn 5549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-mpt 5138  df-cnv 5556  df-dm 5558  df-rn 5559
This theorem is referenced by:  f1omptsnlem  34499  mptsnunlem  34501  dissneqlem  34503
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