Theorem iunopab 4203

Description: Move indexed union inside an ordered-pair abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Assertion

Ref	Expression
iunopab	⊢ ∪ 𝑧 ∈ 𝐴 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 𝜑}

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

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

Proof of Theorem iunopab

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elopab 4180	. . . . 5 ⊢ (𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2	1	rexbii 2442	. . . 4 ⊢ (∃𝑧 ∈ 𝐴 𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑧 ∈ 𝐴 ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
3		rexcom4 2709	. . . . 5 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4		rexcom4 2709	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦∃𝑧 ∈ 𝐴 (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5		r19.42v 2588	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐴 (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑))
6	5	exbii 1584	. . . . . . 7 ⊢ (∃𝑦∃𝑧 ∈ 𝐴 (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑))
7	4, 6	bitri 183	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑))
8	7	exbii 1584	. . . . 5 ⊢ (∃𝑥∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑))
9	3, 8	bitri 183	. . . 4 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑))
10	2, 9	bitri 183	. . 3 ⊢ (∃𝑧 ∈ 𝐴 𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑))
11	10	abbii 2255	. 2 ⊢ {𝑤 ∣ ∃𝑧 ∈ 𝐴 𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑)}
12		df-iun 3815	. 2 ⊢ ∪ 𝑧 ∈ 𝐴 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑧 ∈ 𝐴 𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}}
13		df-opab 3990	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑)}
14	11, 12, 13	3eqtr4i 2170	1 ⊢ ∪ 𝑧 ∈ 𝐴 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 𝜑}

Syntax hints:   ∧ wa 103   = wceq 1331  ∃wex 1468   ∈ wcel 1480  {cab 2125  ∃wrex 2417  ⟨cop 3530  ∪ ciun 3813  {copab 3988

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-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-iun 3815  df-opab 3990

This theorem is referenced by: (None)