Theorem iunrab 5011

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 5010	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜑)} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑)}
2		df-rab 3416	. . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜑)}
3	2	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜑)})
4	3	iuneq2i 4972	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜑)}
5		df-rab 3416	. . 3 ⊢ {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑)}
6		r19.42v 3195	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑))
7	6	abbii 2830	. . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑)}
8	5, 7	eqtr4i 2789	. 2 ⊢ {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑)}
9	1, 4, 8	3eqtr4i 2796	1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝐴 𝜑}

Syntax hints:   ∧ wa 399   = wceq 1561   ∈ wcel 2143  {cab 2741  ∃wrex 3087  {crab 3415  ∪ ciun 4950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1564  df-ex 1801  df-nf 1805  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-ss 3922  df-iun 4952

This theorem is referenced by:  hashrabrex  15854  incexc2  15869  phisum  16827  chnfi  18667  itg2monolem1  25813  aannenlem1  26393  musum  27256  lgsquadlem1  27445  lgsquadlem2  27446  edglnl  29345  rabrexfi  32706  iunpreima  32765  poimirlem27  38147  cnambfre  38168  mapdval3N  42256  mapdval5N  42258  fiphp3d  43397