Theorem iunrab 2022

Description: The indexed union of a restricted class abstraction.

Assertion

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

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

Proof of Theorem iunrab

Step	Hyp	Ref	Expression
1		df-rab 1208	. 2 ⊢ {z ∈ B∣∃x ∈ A [z / y]φ} = {z∣(z ∈ B ∧ ∃x ∈ A [z / y]φ)}
2		ax-17 925	. . 3 ⊢ (x ∈ B → ∀y x ∈ B)
3		ax-17 925	. . 3 ⊢ (x ∈ B → ∀z x ∈ B)
4		ax-17 925	. . 3 ⊢ (∃x ∈ A φ → ∀z∃x ∈ A φ)
5		ax-17 925	. . . 4 ⊢ (x ∈ A → ∀y x ∈ A)
6		hbs1 986	. . . 4 ⊢ ([z / y]φ → ∀y[z / y]φ)
7	5, 6	hbrex 1238	. . 3 ⊢ (∃x ∈ A [z / y]φ → ∀y∃x ∈ A [z / y]φ)
8		sbequ12 865	. . . 4 ⊢ (y = z → (φ ↔ [z / y]φ))
9	8	birexdv 1220	. . 3 ⊢ (y = z → (∃x ∈ A φ ↔ ∃x ∈ A [z / y]φ))
10	2, 3, 4, 7, 9	cbvrab 1425	. 2 ⊢ {y ∈ B∣∃x ∈ A φ} = {z ∈ B∣∃x ∈ A [z / y]φ}
11		eliun 1998	. . . 4 ⊢ (z ∈ ∪x ∈ A {y ∈ B∣φ} ↔ ∃x ∈ A z ∈ {y ∈ B∣φ})
12	2	elrabsf 1456	. . . . 5 ⊢ (z ∈ {y ∈ B∣φ} ↔ (z ∈ B ∧ [z / y]φ))
13	12	birex 1224	. . . 4 ⊢ (∃x ∈ A z ∈ {y ∈ B∣φ} ↔ ∃x ∈ A (z ∈ B ∧ [z / y]φ))
14		r19.42v 1303	. . . 4 ⊢ (∃x ∈ A (z ∈ B ∧ [z / y]φ) ↔ (z ∈ B ∧ ∃x ∈ A [z / y]φ))
15	11, 13, 14	3bitr 155	. . 3 ⊢ (z ∈ ∪x ∈ A {y ∈ B∣φ} ↔ (z ∈ B ∧ ∃x ∈ A [z / y]φ))
16	15	biabri 1180	. 2 ⊢ ∪x ∈ A {y ∈ B∣φ} = {z∣(z ∈ B ∧ ∃x ∈ A [z / y]φ)}
17	1, 10, 16	3eqtr4r 1127	1 ⊢ ∪x ∈ A {y ∈ B∣φ} = {y ∈ B∣∃x ∈ A φ}

Syntax hints:   ∧ wa 196   = weq 797  [wsb 852  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  {crab 1204  ∪ciun 1994

This theorem is referenced by:  iunab 2023

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-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-iun 1996

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