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Theorem coiun1 39875
Description: Composition with an indexed union. Proof analgous to that of coiun 6102. (Contributed by RP, 20-Jun-2020.)
Assertion
Ref Expression
coiun1 ( 𝑥𝐶 𝐴𝐵) = 𝑥𝐶 (𝐴𝐵)
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem coiun1
Dummy variables 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relco 6090 . 2 Rel ( 𝑥𝐶 𝐴𝐵)
2 reliun 5682 . . 3 (Rel 𝑥𝐶 (𝐴𝐵) ↔ ∀𝑥𝐶 Rel (𝐴𝐵))
3 relco 6090 . . . 4 Rel (𝐴𝐵)
43a1i 11 . . 3 (𝑥𝐶 → Rel (𝐴𝐵))
52, 4mprgbir 3150 . 2 Rel 𝑥𝐶 (𝐴𝐵)
6 eliun 4914 . . . . . . . 8 (⟨𝑤, 𝑧⟩ ∈ 𝑥𝐶 𝐴 ↔ ∃𝑥𝐶𝑤, 𝑧⟩ ∈ 𝐴)
7 df-br 5058 . . . . . . . 8 (𝑤 𝑥𝐶 𝐴𝑧 ↔ ⟨𝑤, 𝑧⟩ ∈ 𝑥𝐶 𝐴)
8 df-br 5058 . . . . . . . . 9 (𝑤𝐴𝑧 ↔ ⟨𝑤, 𝑧⟩ ∈ 𝐴)
98rexbii 3244 . . . . . . . 8 (∃𝑥𝐶 𝑤𝐴𝑧 ↔ ∃𝑥𝐶𝑤, 𝑧⟩ ∈ 𝐴)
106, 7, 93bitr4i 304 . . . . . . 7 (𝑤 𝑥𝐶 𝐴𝑧 ↔ ∃𝑥𝐶 𝑤𝐴𝑧)
1110anbi2i 622 . . . . . 6 ((𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧) ↔ (𝑦𝐵𝑤 ∧ ∃𝑥𝐶 𝑤𝐴𝑧))
12 r19.42v 3347 . . . . . 6 (∃𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧) ↔ (𝑦𝐵𝑤 ∧ ∃𝑥𝐶 𝑤𝐴𝑧))
1311, 12bitr4i 279 . . . . 5 ((𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧) ↔ ∃𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
1413exbii 1839 . . . 4 (∃𝑤(𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧) ↔ ∃𝑤𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
15 rexcom4 3246 . . . 4 (∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧) ↔ ∃𝑤𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
1614, 15bitr4i 279 . . 3 (∃𝑤(𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
17 vex 3495 . . . 4 𝑦 ∈ V
18 vex 3495 . . . 4 𝑧 ∈ V
1917, 18opelco 5735 . . 3 (⟨𝑦, 𝑧⟩ ∈ ( 𝑥𝐶 𝐴𝐵) ↔ ∃𝑤(𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧))
20 eliun 4914 . . . 4 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵) ↔ ∃𝑥𝐶𝑦, 𝑧⟩ ∈ (𝐴𝐵))
2117, 18opelco 5735 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2221rexbii 3244 . . . 4 (∃𝑥𝐶𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2320, 22bitri 276 . . 3 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2416, 19, 233bitr4i 304 . 2 (⟨𝑦, 𝑧⟩ ∈ ( 𝑥𝐶 𝐴𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵))
251, 5, 24eqrelriiv 5656 1 ( 𝑥𝐶 𝐴𝐵) = 𝑥𝐶 (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  wex 1771  wcel 2105  wrex 3136  cop 4563   ciun 4910   class class class wbr 5057  ccom 5552  Rel wrel 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-iun 4912  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-co 5557
This theorem is referenced by:  trclfvcom  39946  trclfvdecomr  39951  cotrclrcl  39965
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