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Theorem dfiunv2 4935
 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 4896 . . . 4 𝑦𝐵 𝐶 = {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶}
21a1i 11 . . 3 (𝑥𝐴 𝑦𝐵 𝐶 = {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶})
32iuneq2i 4915 . 2 𝑥𝐴 𝑦𝐵 𝐶 = 𝑥𝐴 {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶}
4 df-iun 4896 . 2 𝑥𝐴 {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶} = {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶}}
5 vex 3472 . . . . 5 𝑧 ∈ V
6 eleq1w 2896 . . . . . 6 (𝑤 = 𝑧 → (𝑤𝐶𝑧𝐶))
76rexbidv 3283 . . . . 5 (𝑤 = 𝑧 → (∃𝑦𝐵 𝑤𝐶 ↔ ∃𝑦𝐵 𝑧𝐶))
85, 7elab 3642 . . . 4 (𝑧 ∈ {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶} ↔ ∃𝑦𝐵 𝑧𝐶)
98rexbii 3235 . . 3 (∃𝑥𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶} ↔ ∃𝑥𝐴𝑦𝐵 𝑧𝐶)
109abbii 2887 . 2 {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶}} = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧𝐶}
113, 4, 103eqtri 2849 1 𝑥𝐴 𝑦𝐵 𝐶 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧𝐶}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2114  {cab 2800  ∃wrex 3131  ∪ ciun 4894 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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794 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 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-in 3915  df-ss 3925  df-iun 4896 This theorem is referenced by:  wspniunwspnon  27707  fusgr2wsp2nb  28117
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