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Theorem coiun 4857
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 4846 . 2 Rel (𝐴 𝑥𝐶 𝐵)
2 reliun 4485 . . 3 (Rel 𝑥𝐶 (𝐴𝐵) ↔ ∀𝑥𝐶 Rel (𝐴𝐵))
3 relco 4846 . . . 4 Rel (𝐴𝐵)
43a1i 9 . . 3 (𝑥𝐶 → Rel (𝐴𝐵))
52, 4mprgbir 2396 . 2 Rel 𝑥𝐶 (𝐴𝐵)
6 eliun 3688 . . . . . . . 8 (⟨𝑦, 𝑤⟩ ∈ 𝑥𝐶 𝐵 ↔ ∃𝑥𝐶𝑦, 𝑤⟩ ∈ 𝐵)
7 df-br 3792 . . . . . . . 8 (𝑦 𝑥𝐶 𝐵𝑤 ↔ ⟨𝑦, 𝑤⟩ ∈ 𝑥𝐶 𝐵)
8 df-br 3792 . . . . . . . . 9 (𝑦𝐵𝑤 ↔ ⟨𝑦, 𝑤⟩ ∈ 𝐵)
98rexbii 2348 . . . . . . . 8 (∃𝑥𝐶 𝑦𝐵𝑤 ↔ ∃𝑥𝐶𝑦, 𝑤⟩ ∈ 𝐵)
106, 7, 93bitr4i 205 . . . . . . 7 (𝑦 𝑥𝐶 𝐵𝑤 ↔ ∃𝑥𝐶 𝑦𝐵𝑤)
1110anbi1i 439 . . . . . 6 ((𝑦 𝑥𝐶 𝐵𝑤𝑤𝐴𝑧) ↔ (∃𝑥𝐶 𝑦𝐵𝑤𝑤𝐴𝑧))
12 r19.41v 2483 . . . . . 6 (∃𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧) ↔ (∃𝑥𝐶 𝑦𝐵𝑤𝑤𝐴𝑧))
1311, 12bitr4i 180 . . . . 5 ((𝑦 𝑥𝐶 𝐵𝑤𝑤𝐴𝑧) ↔ ∃𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
1413exbii 1512 . . . 4 (∃𝑤(𝑦 𝑥𝐶 𝐵𝑤𝑤𝐴𝑧) ↔ ∃𝑤𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
15 rexcom4 2594 . . . 4 (∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧) ↔ ∃𝑤𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
1614, 15bitr4i 180 . . 3 (∃𝑤(𝑦 𝑥𝐶 𝐵𝑤𝑤𝐴𝑧) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
17 vex 2577 . . . 4 𝑦 ∈ V
18 vex 2577 . . . 4 𝑧 ∈ V
1917, 18opelco 4534 . . 3 (⟨𝑦, 𝑧⟩ ∈ (𝐴 𝑥𝐶 𝐵) ↔ ∃𝑤(𝑦 𝑥𝐶 𝐵𝑤𝑤𝐴𝑧))
20 eliun 3688 . . . 4 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵) ↔ ∃𝑥𝐶𝑦, 𝑧⟩ ∈ (𝐴𝐵))
2117, 18opelco 4534 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2221rexbii 2348 . . . 4 (∃𝑥𝐶𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2320, 22bitri 177 . . 3 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2416, 19, 233bitr4i 205 . 2 (⟨𝑦, 𝑧⟩ ∈ (𝐴 𝑥𝐶 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵))
251, 5, 24eqrelriiv 4461 1 (𝐴 𝑥𝐶 𝐵) = 𝑥𝐶 (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1259  wex 1397  wcel 1409  wrex 2324  cop 3405   ciun 3684   class class class wbr 3791  ccom 4376  Rel wrel 4377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-iun 3686  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379  df-co 4381
This theorem is referenced by: (None)
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