MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iunrab Structured version   Visualization version   GIF version

Theorem iunrab 5057
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 5056 . 2 𝑥𝐴 {𝑦 ∣ (𝑦𝐵𝜑)} = {𝑦 ∣ ∃𝑥𝐴 (𝑦𝐵𝜑)}
2 df-rab 3434 . . . 4 {𝑦𝐵𝜑} = {𝑦 ∣ (𝑦𝐵𝜑)}
32a1i 11 . . 3 (𝑥𝐴 → {𝑦𝐵𝜑} = {𝑦 ∣ (𝑦𝐵𝜑)})
43iuneq2i 5018 . 2 𝑥𝐴 {𝑦𝐵𝜑} = 𝑥𝐴 {𝑦 ∣ (𝑦𝐵𝜑)}
5 df-rab 3434 . . 3 {𝑦𝐵 ∣ ∃𝑥𝐴 𝜑} = {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑥𝐴 𝜑)}
6 r19.42v 3189 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝜑) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 𝜑))
76abbii 2807 . . 3 {𝑦 ∣ ∃𝑥𝐴 (𝑦𝐵𝜑)} = {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑥𝐴 𝜑)}
85, 7eqtr4i 2766 . 2 {𝑦𝐵 ∣ ∃𝑥𝐴 𝜑} = {𝑦 ∣ ∃𝑥𝐴 (𝑦𝐵𝜑)}
91, 4, 83eqtr4i 2773 1 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∃𝑥𝐴 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wcel 2106  {cab 2712  wrex 3068  {crab 3433   ciun 4996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-ss 3980  df-iun 4998
This theorem is referenced by:  hashrabrex  15858  incexc2  15871  phisum  16824  itg2monolem1  25800  aannenlem1  26385  musum  27249  lgsquadlem1  27439  lgsquadlem2  27440  edglnl  29175  rabrexfi  32534  iunpreima  32585  poimirlem27  37634  cnambfre  37655  mapdval3N  41614  mapdval5N  41616  fiphp3d  42807
  Copyright terms: Public domain W3C validator