Theorem iunsn 4991

Description: Indexed union of a singleton. Compare dfiun2 4959 and rnmpt 5853. (Contributed by Steven Nguyen, 7-Jun-2023.)

Assertion

Ref	Expression
iunsn	⊢ ∪ 𝑥 ∈ 𝐴 {𝐵} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem iunsn

Step	Hyp	Ref	Expression
1		df-iun 4923	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝐵} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝐵}}
2		velsn 4574	. . . 4 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
3	2	rexbii 3177	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {𝐵} ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
4	3	abbii 2809	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝐵}} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
5	1, 4	eqtri 2766	1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝐵} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}

Syntax hints:   = wceq 1539   ∈ wcel 2108  {cab 2715  ∃wrex 3064  {csn 4558  ∪ ciun 4921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rex 3069  df-v 3424  df-sn 4559  df-iun 4923

This theorem is referenced by:  dfqs3  40139  fsetabsnop  44431