Theorem iunsn 5032

Description: Indexed union of a singleton. Compare dfiun2 4998 and rnmpt 5949. (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 4960	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝐵} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝐵}}
2		velsn 4607	. . . 4 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
3	2	rexbii 3114	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {𝐵} ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
4	3	abbii 2832	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝐵}} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
5	1, 4	eqtri 2788	1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝐵} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  {cab 2743  ∃wrex 3091  {csn 4591  ∪ ciun 4958

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rex 3092  df-v 3459  df-sn 4592  df-iun 4960

This theorem is used by:  pzriprnglem11  21691  dfqs3  43065  fsetabsnop  47845