Theorem rnmptsn 37318

Description: The range of a function mapping to singletons. (Contributed by ML, 15-Jul-2020.)

Assertion

Ref	Expression
rnmptsn	⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}

Distinct variable groups:   𝑢,𝐴   𝑥,𝑢

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem rnmptsn

Step	Hyp	Ref	Expression
1		rnopab 5968	. 2 ⊢ ran {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} = {𝑢 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})}
2		df-mpt 5232	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})}
3	2	rneqi 5951	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = ran {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})}
4		df-rex 3069	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}))
5	4	abbii 2807	. 2 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} = {𝑢 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})}
6	1, 3, 5	3eqtr4i 2773	1 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}

Syntax hints:   ∧ wa 395   = wceq 1537  ∃wex 1776   ∈ wcel 2106  {cab 2712  ∃wrex 3068  {csn 4631  {copab 5210   ↦ cmpt 5231  ran crn 5690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-mpt 5232  df-cnv 5697  df-dm 5699  df-rn 5700

This theorem is referenced by:  f1omptsnlem  37319  mptsnunlem  37321  dissneqlem  37323