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Theorem dfiunv2 4964
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 4926 . . . 4 𝑦𝐵 𝐶 = {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶}
21a1i 11 . . 3 (𝑥𝐴 𝑦𝐵 𝐶 = {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶})
32iuneq2i 4945 . 2 𝑥𝐴 𝑦𝐵 𝐶 = 𝑥𝐴 {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶}
4 df-iun 4926 . 2 𝑥𝐴 {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶} = {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶}}
5 vex 3433 . . . . 5 𝑧 ∈ V
6 eleq1w 2821 . . . . . 6 (𝑤 = 𝑧 → (𝑤𝐶𝑧𝐶))
76rexbidv 3224 . . . . 5 (𝑤 = 𝑧 → (∃𝑦𝐵 𝑤𝐶 ↔ ∃𝑦𝐵 𝑧𝐶))
85, 7elab 3608 . . . 4 (𝑧 ∈ {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶} ↔ ∃𝑦𝐵 𝑧𝐶)
98rexbii 3179 . . 3 (∃𝑥𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶} ↔ ∃𝑥𝐴𝑦𝐵 𝑧𝐶)
109abbii 2808 . 2 {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶}} = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧𝐶}
113, 4, 103eqtri 2770 1 𝑥𝐴 𝑦𝐵 𝐶 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧𝐶}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2106  {cab 2715  wrex 3065   ciun 4924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3431  df-in 3893  df-ss 3903  df-iun 4926
This theorem is referenced by:  wspniunwspnon  28296  fusgr2wsp2nb  28706
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