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Theorem dfiunv2 4708
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 4674 . . . 4 𝑦𝐵 𝐶 = {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶}
21a1i 11 . . 3 (𝑥𝐴 𝑦𝐵 𝐶 = {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶})
32iuneq2i 4691 . 2 𝑥𝐴 𝑦𝐵 𝐶 = 𝑥𝐴 {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶}
4 df-iun 4674 . 2 𝑥𝐴 {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶} = {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶}}
5 vex 3343 . . . . 5 𝑧 ∈ V
6 eleq1w 2822 . . . . . 6 (𝑤 = 𝑧 → (𝑤𝐶𝑧𝐶))
76rexbidv 3190 . . . . 5 (𝑤 = 𝑧 → (∃𝑦𝐵 𝑤𝐶 ↔ ∃𝑦𝐵 𝑧𝐶))
85, 7elab 3490 . . . 4 (𝑧 ∈ {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶} ↔ ∃𝑦𝐵 𝑧𝐶)
98rexbii 3179 . . 3 (∃𝑥𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶} ↔ ∃𝑥𝐴𝑦𝐵 𝑧𝐶)
109abbii 2877 . 2 {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦𝐵 𝑤𝐶}} = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧𝐶}
113, 4, 103eqtri 2786 1 𝑥𝐴 𝑦𝐵 𝐶 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧𝐶}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  wcel 2139  {cab 2746  wrex 3051   ciun 4672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-in 3722  df-ss 3729  df-iun 4674
This theorem is referenced by:  wspniunwspnon  27064  fusgr2wsp2nb  27509
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