Theorem rnmptsn 37651

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 5909	. 2 ⊢ ran {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} = {𝑢 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})}
2		df-mpt 5167	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})}
3	2	rneqi 5892	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = ran {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})}
4		df-rex 3062	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥}))
5	4	abbii 2803	. 2 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} = {𝑢 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})}
6	1, 3, 5	3eqtr4i 2769	1 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}

Syntax hints:   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2714  ∃wrex 3061  {csn 4567  {copab 5147   ↦ cmpt 5166  ran crn 5632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-mpt 5167  df-cnv 5639  df-dm 5641  df-rn 5642

This theorem is referenced by:  f1omptsnlem  37652  mptsnunlem  37654  dissneqlem  37656