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Theorem coiun 6090
Description: Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)
Assertion
Ref Expression
coiun (𝐴 𝑥𝐶 𝐵) = 𝑥𝐶 (𝐴𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem coiun
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relco 6078 . 2 Rel (𝐴 𝑥𝐶 𝐵)
2 reliun 5670 . . 3 (Rel 𝑥𝐶 (𝐴𝐵) ↔ ∀𝑥𝐶 Rel (𝐴𝐵))
3 relco 6078 . . . 4 Rel (𝐴𝐵)
43a1i 11 . . 3 (𝑥𝐶 → Rel (𝐴𝐵))
52, 4mprgbir 3147 . 2 Rel 𝑥𝐶 (𝐴𝐵)
6 eliun 4904 . . . . . . . . 9 (⟨𝑦, 𝑤⟩ ∈ 𝑥𝐶 𝐵 ↔ ∃𝑥𝐶𝑦, 𝑤⟩ ∈ 𝐵)
7 df-br 5048 . . . . . . . . 9 (𝑦 𝑥𝐶 𝐵𝑤 ↔ ⟨𝑦, 𝑤⟩ ∈ 𝑥𝐶 𝐵)
8 df-br 5048 . . . . . . . . . 10 (𝑦𝐵𝑤 ↔ ⟨𝑦, 𝑤⟩ ∈ 𝐵)
98rexbii 3241 . . . . . . . . 9 (∃𝑥𝐶 𝑦𝐵𝑤 ↔ ∃𝑥𝐶𝑦, 𝑤⟩ ∈ 𝐵)
106, 7, 93bitr4i 306 . . . . . . . 8 (𝑦 𝑥𝐶 𝐵𝑤 ↔ ∃𝑥𝐶 𝑦𝐵𝑤)
1110anbi1i 626 . . . . . . 7 ((𝑦 𝑥𝐶 𝐵𝑤𝑤𝐴𝑧) ↔ (∃𝑥𝐶 𝑦𝐵𝑤𝑤𝐴𝑧))
12 r19.41v 3338 . . . . . . 7 (∃𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧) ↔ (∃𝑥𝐶 𝑦𝐵𝑤𝑤𝐴𝑧))
1311, 12bitr4i 281 . . . . . 6 ((𝑦 𝑥𝐶 𝐵𝑤𝑤𝐴𝑧) ↔ ∃𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
1413exbii 1849 . . . . 5 (∃𝑤(𝑦 𝑥𝐶 𝐵𝑤𝑤𝐴𝑧) ↔ ∃𝑤𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
15 rexcom4 3243 . . . . 5 (∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧) ↔ ∃𝑤𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
1614, 15bitr4i 281 . . . 4 (∃𝑤(𝑦 𝑥𝐶 𝐵𝑤𝑤𝐴𝑧) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
17 vex 3482 . . . . 5 𝑦 ∈ V
18 vex 3482 . . . . 5 𝑧 ∈ V
1917, 18opelco 5723 . . . 4 (⟨𝑦, 𝑧⟩ ∈ (𝐴 𝑥𝐶 𝐵) ↔ ∃𝑤(𝑦 𝑥𝐶 𝐵𝑤𝑤𝐴𝑧))
2017, 18opelco 5723 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2120rexbii 3241 . . . 4 (∃𝑥𝐶𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2216, 19, 213bitr4i 306 . . 3 (⟨𝑦, 𝑧⟩ ∈ (𝐴 𝑥𝐶 𝐵) ↔ ∃𝑥𝐶𝑦, 𝑧⟩ ∈ (𝐴𝐵))
23 eliun 4904 . . 3 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵) ↔ ∃𝑥𝐶𝑦, 𝑧⟩ ∈ (𝐴𝐵))
2422, 23bitr4i 281 . 2 (⟨𝑦, 𝑧⟩ ∈ (𝐴 𝑥𝐶 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵))
251, 5, 24eqrelriiv 5644 1 (𝐴 𝑥𝐶 𝐵) = 𝑥𝐶 (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wex 1781  wcel 2115  wrex 3133  cop 4554   ciun 4900   class class class wbr 5047  ccom 5540  Rel wrel 5541
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-iun 4902  df-br 5048  df-opab 5110  df-xp 5542  df-rel 5543  df-co 5545
This theorem is referenced by:  fparlem3  7792  fparlem4  7793  trclrelexplem  40244  trclfvcom  40256
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