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Theorem iunab 4939
 Description: The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)
Assertion
Ref Expression
iunab 𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∃𝑥𝐴 𝜑}
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem iunab
StepHypRef Expression
1 nfcv 2955 . . . 4 𝑦𝐴
2 nfab1 2957 . . . 4 𝑦{𝑦𝜑}
31, 2nfiun 4912 . . 3 𝑦 𝑥𝐴 {𝑦𝜑}
4 nfab1 2957 . . 3 𝑦{𝑦 ∣ ∃𝑥𝐴 𝜑}
53, 4cleqf 2983 . 2 ( 𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ ∀𝑦(𝑦 𝑥𝐴 {𝑦𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑}))
6 abid 2780 . . . 4 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
76rexbii 3210 . . 3 (∃𝑥𝐴 𝑦 ∈ {𝑦𝜑} ↔ ∃𝑥𝐴 𝜑)
8 eliun 4886 . . 3 (𝑦 𝑥𝐴 {𝑦𝜑} ↔ ∃𝑥𝐴 𝑦 ∈ {𝑦𝜑})
9 abid 2780 . . 3 (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ ∃𝑥𝐴 𝜑)
107, 8, 93bitr4i 306 . 2 (𝑦 𝑥𝐴 {𝑦𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑})
115, 10mpgbir 1801 1 𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∃𝑥𝐴 𝜑}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  {cab 2776  ∃wrex 3107  ∪ ciun 4882 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-iun 4884 This theorem is referenced by:  iunrab  4940  iunid  4948  dfimafn2  6705  rabiun  35049  dfaimafn2  43765  rnfdmpr  43880
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