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Theorem coiun 5133
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 5122 . 2 Rel (𝐴 𝑥𝐶 𝐵)
2 reliun 4743 . . 3 (Rel 𝑥𝐶 (𝐴𝐵) ↔ ∀𝑥𝐶 Rel (𝐴𝐵))
3 relco 5122 . . . 4 Rel (𝐴𝐵)
43a1i 9 . . 3 (𝑥𝐶 → Rel (𝐴𝐵))
52, 4mprgbir 2535 . 2 Rel 𝑥𝐶 (𝐴𝐵)
6 eliun 3888 . . . . . . . 8 (⟨𝑦, 𝑤⟩ ∈ 𝑥𝐶 𝐵 ↔ ∃𝑥𝐶𝑦, 𝑤⟩ ∈ 𝐵)
7 df-br 4001 . . . . . . . 8 (𝑦 𝑥𝐶 𝐵𝑤 ↔ ⟨𝑦, 𝑤⟩ ∈ 𝑥𝐶 𝐵)
8 df-br 4001 . . . . . . . . 9 (𝑦𝐵𝑤 ↔ ⟨𝑦, 𝑤⟩ ∈ 𝐵)
98rexbii 2484 . . . . . . . 8 (∃𝑥𝐶 𝑦𝐵𝑤 ↔ ∃𝑥𝐶𝑦, 𝑤⟩ ∈ 𝐵)
106, 7, 93bitr4i 212 . . . . . . 7 (𝑦 𝑥𝐶 𝐵𝑤 ↔ ∃𝑥𝐶 𝑦𝐵𝑤)
1110anbi1i 458 . . . . . 6 ((𝑦 𝑥𝐶 𝐵𝑤𝑤𝐴𝑧) ↔ (∃𝑥𝐶 𝑦𝐵𝑤𝑤𝐴𝑧))
12 r19.41v 2633 . . . . . 6 (∃𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧) ↔ (∃𝑥𝐶 𝑦𝐵𝑤𝑤𝐴𝑧))
1311, 12bitr4i 187 . . . . 5 ((𝑦 𝑥𝐶 𝐵𝑤𝑤𝐴𝑧) ↔ ∃𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
1413exbii 1605 . . . 4 (∃𝑤(𝑦 𝑥𝐶 𝐵𝑤𝑤𝐴𝑧) ↔ ∃𝑤𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
15 rexcom4 2760 . . . 4 (∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧) ↔ ∃𝑤𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
1614, 15bitr4i 187 . . 3 (∃𝑤(𝑦 𝑥𝐶 𝐵𝑤𝑤𝐴𝑧) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
17 vex 2740 . . . 4 𝑦 ∈ V
18 vex 2740 . . . 4 𝑧 ∈ V
1917, 18opelco 4794 . . 3 (⟨𝑦, 𝑧⟩ ∈ (𝐴 𝑥𝐶 𝐵) ↔ ∃𝑤(𝑦 𝑥𝐶 𝐵𝑤𝑤𝐴𝑧))
20 eliun 3888 . . . 4 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵) ↔ ∃𝑥𝐶𝑦, 𝑧⟩ ∈ (𝐴𝐵))
2117, 18opelco 4794 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2221rexbii 2484 . . . 4 (∃𝑥𝐶𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2320, 22bitri 184 . . 3 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2416, 19, 233bitr4i 212 . 2 (⟨𝑦, 𝑧⟩ ∈ (𝐴 𝑥𝐶 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵))
251, 5, 24eqrelriiv 4716 1 (𝐴 𝑥𝐶 𝐵) = 𝑥𝐶 (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  wex 1492  wcel 2148  wrex 2456  cop 3594   ciun 3884   class class class wbr 4000  ccom 4626  Rel wrel 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-iun 3886  df-br 4001  df-opab 4062  df-xp 4628  df-rel 4629  df-co 4631
This theorem is referenced by: (None)
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