Theorem iunopab 4312

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 4288	. . . . 5 ⊢ (𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2	1	rexbii 2501	. . . 4 ⊢ (∃𝑧 ∈ 𝐴 𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑧 ∈ 𝐴 ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
3		rexcom4 2783	. . . . 5 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4		rexcom4 2783	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦∃𝑧 ∈ 𝐴 (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5		r19.42v 2651	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐴 (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑))
6	5	exbii 1616	. . . . . . 7 ⊢ (∃𝑦∃𝑧 ∈ 𝐴 (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑))
7	4, 6	bitri 184	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑))
8	7	exbii 1616	. . . . 5 ⊢ (∃𝑥∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑))
9	3, 8	bitri 184	. . . 4 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑))
10	2, 9	bitri 184	. . 3 ⊢ (∃𝑧 ∈ 𝐴 𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑))
11	10	abbii 2309	. 2 ⊢ {𝑤 ∣ ∃𝑧 ∈ 𝐴 𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑)}
12		df-iun 3914	. 2 ⊢ ∪ 𝑧 ∈ 𝐴 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑧 ∈ 𝐴 𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}}
13		df-opab 4091	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧 ∈ 𝐴 𝜑)}
14	11, 12, 13	3eqtr4i 2224	1 ⊢ ∪ 𝑧 ∈ 𝐴 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 𝜑}

Syntax hints:   ∧ wa 104   = wceq 1364  ∃wex 1503   ∈ wcel 2164  {cab 2179  ∃wrex 2473  ⟨cop 3621  ∪ ciun 3912  {copab 4089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-iun 3914  df-opab 4091

This theorem is referenced by: (None)