Theorem rnmptsn 38449

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 5942	. 2 ⊢ ran {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} = {𝑢 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})}
2		df-mpt 5190	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})}
3	2	rneqi 5925	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = ran {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})}
4		df-rex 3091	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}))
5	4	abbii 2830	. 2 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} = {𝑢 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})}
6	1, 3, 5	3eqtr4i 2796	1 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∃wrex 3090  {csn 4588  {copab 5170   ↦ cmpt 5189  ran crn 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-mpt 5190  df-cnv 5663  df-dm 5665  df-rn 5666

This theorem is used by:  f1omptsnlem  38450  mptsnunlem  38452  dissneqlem  38454