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Theorem dfiunv2 4527
 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 4492 . . . 4 𝑦𝐵 𝐶 = {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶}
21a1i 11 . . 3 (𝑥𝐴 𝑦𝐵 𝐶 = {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶})
32iuneq2i 4510 . 2 𝑥𝐴 𝑦𝐵 𝐶 = 𝑥𝐴 {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶}
4 df-iun 4492 . 2 𝑥𝐴 {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶} = {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶}}
5 vex 3192 . . . . 5 𝑧 ∈ V
6 eleq1 2686 . . . . . 6 (𝑤 = 𝑧 → (𝑤𝐶𝑧𝐶))
76rexbidv 3046 . . . . 5 (𝑤 = 𝑧 → (∃𝑦𝐵 𝑤𝐶 ↔ ∃𝑦𝐵 𝑧𝐶))
85, 7elab 3337 . . . 4 (𝑧 ∈ {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶} ↔ ∃𝑦𝐵 𝑧𝐶)
98rexbii 3035 . . 3 (∃𝑥𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶} ↔ ∃𝑥𝐴𝑦𝐵 𝑧𝐶)
109abbii 2736 . 2 {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶}} = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧𝐶}
113, 4, 103eqtri 2647 1 𝑥𝐴 𝑦𝐵 𝐶 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧𝐶}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1480   ∈ wcel 1987  {cab 2607  ∃wrex 2908  ∪ ciun 4490 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3191  df-in 3566  df-ss 3573  df-iun 4492 This theorem is referenced by:  wspniunwspnon  26705  fusgr2wsp2nb  27073
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