Theorem iunrab 3860

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 3859	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜑)} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑)}
2		df-rab 2425	. . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜑)}
3	2	a1i 9	. . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜑)})
4	3	iuneq2i 3831	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜑)}
5		df-rab 2425	. . 3 ⊢ {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑)}
6		r19.42v 2588	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑))
7	6	abbii 2255	. . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑)}
8	5, 7	eqtr4i 2163	. 2 ⊢ {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑)}
9	1, 4, 8	3eqtr4i 2170	1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝐴 𝜑}

Syntax hints:   ∧ wa 103   = wceq 1331   ∈ wcel 1480  {cab 2125  ∃wrex 2417  {crab 2420  ∪ ciun 3813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-in 3077  df-ss 3084  df-iun 3815

This theorem is referenced by:  hashrabrex  11250