ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iunrab GIF version

Theorem iunrab 3772
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 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∃𝑥𝐴 𝜑}
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem iunrab
StepHypRef Expression
1 iunab 3771 . 2 𝑥𝐴 {𝑦 ∣ (𝑦𝐵𝜑)} = {𝑦 ∣ ∃𝑥𝐴 (𝑦𝐵𝜑)}
2 df-rab 2368 . . . 4 {𝑦𝐵𝜑} = {𝑦 ∣ (𝑦𝐵𝜑)}
32a1i 9 . . 3 (𝑥𝐴 → {𝑦𝐵𝜑} = {𝑦 ∣ (𝑦𝐵𝜑)})
43iuneq2i 3743 . 2 𝑥𝐴 {𝑦𝐵𝜑} = 𝑥𝐴 {𝑦 ∣ (𝑦𝐵𝜑)}
5 df-rab 2368 . . 3 {𝑦𝐵 ∣ ∃𝑥𝐴 𝜑} = {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑥𝐴 𝜑)}
6 r19.42v 2524 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝜑) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 𝜑))
76abbii 2203 . . 3 {𝑦 ∣ ∃𝑥𝐴 (𝑦𝐵𝜑)} = {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑥𝐴 𝜑)}
85, 7eqtr4i 2111 . 2 {𝑦𝐵 ∣ ∃𝑥𝐴 𝜑} = {𝑦 ∣ ∃𝑥𝐴 (𝑦𝐵𝜑)}
91, 4, 83eqtr4i 2118 1 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∃𝑥𝐴 𝜑}
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1289  wcel 1438  {cab 2074  wrex 2360  {crab 2363   ciun 3725
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-in 3003  df-ss 3010  df-iun 3727
This theorem is referenced by:  hashrabrex  10837
  Copyright terms: Public domain W3C validator