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Theorem rnmptsn 35118
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 5791 . 2 ran {⟨𝑥, 𝑢⟩ ∣ (𝑥𝐴𝑢 = {𝑥})} = {𝑢 ∣ ∃𝑥(𝑥𝐴𝑢 = {𝑥})}
2 df-mpt 5108 . . 3 (𝑥𝐴 ↦ {𝑥}) = {⟨𝑥, 𝑢⟩ ∣ (𝑥𝐴𝑢 = {𝑥})}
32rneqi 5774 . 2 ran (𝑥𝐴 ↦ {𝑥}) = ran {⟨𝑥, 𝑢⟩ ∣ (𝑥𝐴𝑢 = {𝑥})}
4 df-rex 3059 . . 3 (∃𝑥𝐴 𝑢 = {𝑥} ↔ ∃𝑥(𝑥𝐴𝑢 = {𝑥}))
54abbii 2803 . 2 {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}} = {𝑢 ∣ ∃𝑥(𝑥𝐴𝑢 = {𝑥})}
61, 3, 53eqtr4i 2771 1 ran (𝑥𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1542  wex 1786  wcel 2113  {cab 2716  wrex 3054  {csn 4513  {copab 5089  cmpt 5107  ran crn 5520
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-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-rex 3059  df-v 3399  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-br 5028  df-opab 5090  df-mpt 5108  df-cnv 5527  df-dm 5529  df-rn 5530
This theorem is referenced by:  f1omptsnlem  35119  mptsnunlem  35121  dissneqlem  35123
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