Theorem iunab 2023

Description: The indexed union of a class abstraction.

Assertion

Ref	Expression
iunab	⊢ ∪x ∈ A {y∣φ} = {y∣∃x ∈ A φ}

Distinct variable group(s):   y,A   x,y

Proof of Theorem iunab

Step	Hyp	Ref	Expression
1		iunrab 2022	. 2 ⊢ ∪x ∈ A {y ∈ V∣φ} = {y ∈ V∣∃x ∈ A φ}
2		rabab 1359	. . . 4 ⊢ {y ∈ V∣φ} = {y∣φ}
3	2	a1i 7	. . 3 ⊢ (x ∈ A → {y ∈ V∣φ} = {y∣φ})
4	3	iuneq2i 2008	. 2 ⊢ ∪x ∈ A {y ∈ V∣φ} = ∪x ∈ A {y∣φ}
5		rabab 1359	. 2 ⊢ {y ∈ V∣∃x ∈ A φ} = {y∣∃x ∈ A φ}
6	1, 4, 5	3eqtr3 1124	1 ⊢ ∪x ∈ A {y∣φ} = {y∣∃x ∈ A φ}

Syntax hints:  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  {crab 1204  Vcvv 1348  ∪ciun 1994

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-in 1491  df-ss 1492  df-iun 1996

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