Theorem iunsn 5030

Description: Indexed union of a singleton. Compare dfiun2 4997 and rnmpt 5921. (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 4957	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝐵} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝐵}}
2		velsn 4605	. . . 4 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
3	2	rexbii 3076	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {𝐵} ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
4	3	abbii 2796	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝐵}} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
5	1, 4	eqtri 2752	1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝐵} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}

Syntax hints:   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053  {csn 4589  ∪ ciun 4955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rex 3054  df-v 3449  df-sn 4590  df-iun 4957

This theorem is referenced by:  pzriprnglem11  21401  dfqs3  42226  fsetabsnop  47051