Theorem iunrab 4013

Description: The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)

Assertion

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

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

Allowed substitution hints:   φ(x,y)   A(x)   B(y)

Proof of Theorem iunrab

Step	Hyp	Ref	Expression
1		iunab 4012	. 2 ⊢ ∪x ∈ A {y ∣ (y ∈ B ∧ φ)} = {y ∣ ∃x ∈ A (y ∈ B ∧ φ)}
2		df-rab 2623	. . . 4 ⊢ {y ∈ B ∣ φ} = {y ∣ (y ∈ B ∧ φ)}
3	2	a1i 10	. . 3 ⊢ (x ∈ A → {y ∈ B ∣ φ} = {y ∣ (y ∈ B ∧ φ)})
4	3	iuneq2i 3987	. 2 ⊢ ∪x ∈ A {y ∈ B ∣ φ} = ∪x ∈ A {y ∣ (y ∈ B ∧ φ)}
5		df-rab 2623	. . 3 ⊢ {y ∈ B ∣ ∃x ∈ A φ} = {y ∣ (y ∈ B ∧ ∃x ∈ A φ)}
6		r19.42v 2765	. . . 4 ⊢ (∃x ∈ A (y ∈ B ∧ φ) ↔ (y ∈ B ∧ ∃x ∈ A φ))
7	6	abbii 2465	. . 3 ⊢ {y ∣ ∃x ∈ A (y ∈ B ∧ φ)} = {y ∣ (y ∈ B ∧ ∃x ∈ A φ)}
8	5, 7	eqtr4i 2376	. 2 ⊢ {y ∈ B ∣ ∃x ∈ A φ} = {y ∣ ∃x ∈ A (y ∈ B ∧ φ)}
9	1, 4, 8	3eqtr4i 2383	1 ⊢ ∪x ∈ A {y ∈ B ∣ φ} = {y ∈ B ∣ ∃x ∈ A φ}

Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  {crab 2618  ∪ciun 3969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-iun 3971

This theorem is referenced by: (None)