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Theorem iunrab 4967
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 4966 . 2 𝑥𝐴 {𝑦 ∣ (𝑦𝐵𝜑)} = {𝑦 ∣ ∃𝑥𝐴 (𝑦𝐵𝜑)}
2 df-rab 3145 . . . 4 {𝑦𝐵𝜑} = {𝑦 ∣ (𝑦𝐵𝜑)}
32a1i 11 . . 3 (𝑥𝐴 → {𝑦𝐵𝜑} = {𝑦 ∣ (𝑦𝐵𝜑)})
43iuneq2i 4931 . 2 𝑥𝐴 {𝑦𝐵𝜑} = 𝑥𝐴 {𝑦 ∣ (𝑦𝐵𝜑)}
5 df-rab 3145 . . 3 {𝑦𝐵 ∣ ∃𝑥𝐴 𝜑} = {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑥𝐴 𝜑)}
6 r19.42v 3348 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝜑) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 𝜑))
76abbii 2884 . . 3 {𝑦 ∣ ∃𝑥𝐴 (𝑦𝐵𝜑)} = {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑥𝐴 𝜑)}
85, 7eqtr4i 2845 . 2 {𝑦𝐵 ∣ ∃𝑥𝐴 𝜑} = {𝑦 ∣ ∃𝑥𝐴 (𝑦𝐵𝜑)}
91, 4, 83eqtr4i 2852 1 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∃𝑥𝐴 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 398   = wceq 1530  wcel 2107  {cab 2797  wrex 3137  {crab 3140   ciun 4910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-in 3941  df-ss 3950  df-iun 4912
This theorem is referenced by:  hashrabrex  15172  incexc2  15185  phisum  16119  itg2monolem1  24343  aannenlem1  24909  musum  25760  lgsquadlem1  25948  lgsquadlem2  25949  edglnl  26920  iunpreima  30308  poimirlem27  34901  cnambfre  34922  mapdval3N  38749  mapdval5N  38751  fiphp3d  39396
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