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Theorem coiun 5807
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 5795 . 2 Rel (𝐴 𝑥𝐶 𝐵)
2 reliun 5396 . . 3 (Rel 𝑥𝐶 (𝐴𝐵) ↔ ∀𝑥𝐶 Rel (𝐴𝐵))
3 relco 5795 . . . 4 Rel (𝐴𝐵)
43a1i 11 . . 3 (𝑥𝐶 → Rel (𝐴𝐵))
52, 4mprgbir 3066 . 2 Rel 𝑥𝐶 (𝐴𝐵)
6 eliun 4677 . . . . . . . . 9 (⟨𝑦, 𝑤⟩ ∈ 𝑥𝐶 𝐵 ↔ ∃𝑥𝐶𝑦, 𝑤⟩ ∈ 𝐵)
7 df-br 4806 . . . . . . . . 9 (𝑦 𝑥𝐶 𝐵𝑤 ↔ ⟨𝑦, 𝑤⟩ ∈ 𝑥𝐶 𝐵)
8 df-br 4806 . . . . . . . . . 10 (𝑦𝐵𝑤 ↔ ⟨𝑦, 𝑤⟩ ∈ 𝐵)
98rexbii 3180 . . . . . . . . 9 (∃𝑥𝐶 𝑦𝐵𝑤 ↔ ∃𝑥𝐶𝑦, 𝑤⟩ ∈ 𝐵)
106, 7, 93bitr4i 292 . . . . . . . 8 (𝑦 𝑥𝐶 𝐵𝑤 ↔ ∃𝑥𝐶 𝑦𝐵𝑤)
1110anbi1i 733 . . . . . . 7 ((𝑦 𝑥𝐶 𝐵𝑤𝑤𝐴𝑧) ↔ (∃𝑥𝐶 𝑦𝐵𝑤𝑤𝐴𝑧))
12 r19.41v 3228 . . . . . . 7 (∃𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧) ↔ (∃𝑥𝐶 𝑦𝐵𝑤𝑤𝐴𝑧))
1311, 12bitr4i 267 . . . . . 6 ((𝑦 𝑥𝐶 𝐵𝑤𝑤𝐴𝑧) ↔ ∃𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
1413exbii 1923 . . . . 5 (∃𝑤(𝑦 𝑥𝐶 𝐵𝑤𝑤𝐴𝑧) ↔ ∃𝑤𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
15 rexcom4 3366 . . . . 5 (∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧) ↔ ∃𝑤𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
1614, 15bitr4i 267 . . . 4 (∃𝑤(𝑦 𝑥𝐶 𝐵𝑤𝑤𝐴𝑧) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
17 vex 3344 . . . . 5 𝑦 ∈ V
18 vex 3344 . . . . 5 𝑧 ∈ V
1917, 18opelco 5450 . . . 4 (⟨𝑦, 𝑧⟩ ∈ (𝐴 𝑥𝐶 𝐵) ↔ ∃𝑤(𝑦 𝑥𝐶 𝐵𝑤𝑤𝐴𝑧))
2017, 18opelco 5450 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2120rexbii 3180 . . . 4 (∃𝑥𝐶𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2216, 19, 213bitr4i 292 . . 3 (⟨𝑦, 𝑧⟩ ∈ (𝐴 𝑥𝐶 𝐵) ↔ ∃𝑥𝐶𝑦, 𝑧⟩ ∈ (𝐴𝐵))
23 eliun 4677 . . 3 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵) ↔ ∃𝑥𝐶𝑦, 𝑧⟩ ∈ (𝐴𝐵))
2422, 23bitr4i 267 . 2 (⟨𝑦, 𝑧⟩ ∈ (𝐴 𝑥𝐶 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵))
251, 5, 24eqrelriiv 5372 1 (𝐴 𝑥𝐶 𝐵) = 𝑥𝐶 (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wex 1853  wcel 2140  wrex 3052  cop 4328   ciun 4673   class class class wbr 4805  ccom 5271  Rel wrel 5272
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 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-iun 4675  df-br 4806  df-opab 4866  df-xp 5273  df-rel 5274  df-co 5276
This theorem is referenced by:  fparlem3  7449  fparlem4  7450  trclrelexplem  38524  trclfvcom  38536
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