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Theorem rnmptsn 37337
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 5964 . 2 ran {⟨𝑥, 𝑢⟩ ∣ (𝑥𝐴𝑢 = {𝑥})} = {𝑢 ∣ ∃𝑥(𝑥𝐴𝑢 = {𝑥})}
2 df-mpt 5225 . . 3 (𝑥𝐴 ↦ {𝑥}) = {⟨𝑥, 𝑢⟩ ∣ (𝑥𝐴𝑢 = {𝑥})}
32rneqi 5947 . 2 ran (𝑥𝐴 ↦ {𝑥}) = ran {⟨𝑥, 𝑢⟩ ∣ (𝑥𝐴𝑢 = {𝑥})}
4 df-rex 3070 . . 3 (∃𝑥𝐴 𝑢 = {𝑥} ↔ ∃𝑥(𝑥𝐴𝑢 = {𝑥}))
54abbii 2808 . 2 {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}} = {𝑢 ∣ ∃𝑥(𝑥𝐴𝑢 = {𝑥})}
61, 3, 53eqtr4i 2774 1 ran (𝑥𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1539  wex 1778  wcel 2107  {cab 2713  wrex 3069  {csn 4625  {copab 5204  cmpt 5224  ran crn 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-mpt 5225  df-cnv 5692  df-dm 5694  df-rn 5695
This theorem is referenced by:  f1omptsnlem  37338  mptsnunlem  37340  dissneqlem  37342
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