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Theorem iunab 3731
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 2194 . . . 4 𝑦𝐴
2 nfab1 2196 . . . 4 𝑦{𝑦𝜑}
31, 2nfiunxy 3711 . . 3 𝑦 𝑥𝐴 {𝑦𝜑}
4 nfab1 2196 . . 3 𝑦{𝑦 ∣ ∃𝑥𝐴 𝜑}
53, 4cleqf 2217 . 2 ( 𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ ∀𝑦(𝑦 𝑥𝐴 {𝑦𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑}))
6 abid 2044 . . . 4 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
76rexbii 2348 . . 3 (∃𝑥𝐴 𝑦 ∈ {𝑦𝜑} ↔ ∃𝑥𝐴 𝜑)
8 eliun 3689 . . 3 (𝑦 𝑥𝐴 {𝑦𝜑} ↔ ∃𝑥𝐴 𝑦 ∈ {𝑦𝜑})
9 abid 2044 . . 3 (𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑} ↔ ∃𝑥𝐴 𝜑)
107, 8, 93bitr4i 205 . 2 (𝑦 𝑥𝐴 {𝑦𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥𝐴 𝜑})
115, 10mpgbir 1358 1 𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∃𝑥𝐴 𝜑}
Colors of variables: wff set class
Syntax hints:  wb 102   = wceq 1259  wcel 1409  {cab 2042  wrex 2324   ciun 3685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-iun 3687
This theorem is referenced by:  iunrab  3732  iunid  3740  dfimafn2  5251
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