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Theorem rnmptsn 33312
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 df-mpt 4763 . . . 4 (𝑥𝐴 ↦ {𝑥}) = {⟨𝑥, 𝑢⟩ ∣ (𝑥𝐴𝑢 = {𝑥})}
21rneqi 5384 . . 3 ran (𝑥𝐴 ↦ {𝑥}) = ran {⟨𝑥, 𝑢⟩ ∣ (𝑥𝐴𝑢 = {𝑥})}
3 rnopab 5402 . . 3 ran {⟨𝑥, 𝑢⟩ ∣ (𝑥𝐴𝑢 = {𝑥})} = {𝑢 ∣ ∃𝑥(𝑥𝐴𝑢 = {𝑥})}
42, 3eqtri 2673 . 2 ran (𝑥𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥(𝑥𝐴𝑢 = {𝑥})}
5 df-rex 2947 . . 3 (∃𝑥𝐴 𝑢 = {𝑥} ↔ ∃𝑥(𝑥𝐴𝑢 = {𝑥}))
65abbii 2768 . 2 {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}} = {𝑢 ∣ ∃𝑥(𝑥𝐴𝑢 = {𝑥})}
74, 6eqtr4i 2676 1 ran (𝑥𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wrex 2942  {csn 4210  {copab 4745  cmpt 4762  ran crn 5144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-mpt 4763  df-cnv 5151  df-dm 5153  df-rn 5154
This theorem is referenced by:  f1omptsnlem  33313  mptsnunlem  33315  dissneqlem  33317
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