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Theorem dfiunv2 3853
Description: Define double indexed union. (Contributed by FL, 6-Nov-2013.)
Assertion
Ref Expression
dfiunv2 𝑥𝐴 𝑦𝐵 𝐶 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧𝐶}
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧   𝑧,𝐴   𝑧,𝐵   𝑧,𝐶
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem dfiunv2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 df-iun 3819 . . . 4 𝑦𝐵 𝐶 = {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶}
21a1i 9 . . 3 (𝑥𝐴 𝑦𝐵 𝐶 = {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶})
32iuneq2i 3835 . 2 𝑥𝐴 𝑦𝐵 𝐶 = 𝑥𝐴 {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶}
4 df-iun 3819 . 2 𝑥𝐴 {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶} = {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶}}
5 vex 2690 . . . . 5 𝑧 ∈ V
6 eleq1 2203 . . . . . 6 (𝑤 = 𝑧 → (𝑤𝐶𝑧𝐶))
76rexbidv 2439 . . . . 5 (𝑤 = 𝑧 → (∃𝑦𝐵 𝑤𝐶 ↔ ∃𝑦𝐵 𝑧𝐶))
85, 7elab 2829 . . . 4 (𝑧 ∈ {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶} ↔ ∃𝑦𝐵 𝑧𝐶)
98rexbii 2443 . . 3 (∃𝑥𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶} ↔ ∃𝑥𝐴𝑦𝐵 𝑧𝐶)
109abbii 2256 . 2 {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶}} = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧𝐶}
113, 4, 103eqtri 2165 1 𝑥𝐴 𝑦𝐵 𝐶 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧𝐶}
Colors of variables: wff set class
Syntax hints:   = wceq 1332  wcel 1481  {cab 2126  wrex 2418   ciun 3817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689  df-in 3078  df-ss 3085  df-iun 3819
This theorem is referenced by: (None)
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