Theorem rnmptsn 37379

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 5893	. 2 ⊢ ran {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} = {𝑢 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})}
2		df-mpt 5171	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})}
3	2	rneqi 5876	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = ran {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})}
4		df-rex 3057	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}))
5	4	abbii 2798	. 2 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} = {𝑢 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})}
6	1, 3, 5	3eqtr4i 2764	1 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}

Syntax hints:   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  ∃wrex 3056  {csn 4573  {copab 5151   ↦ cmpt 5170  ran crn 5615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-mpt 5171  df-cnv 5622  df-dm 5624  df-rn 5625

This theorem is referenced by:  f1omptsnlem  37380  mptsnunlem  37382  dissneqlem  37384