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Theorem coiun1 43641
Description: Composition with an indexed union. Proof analogous to that of coiun 6277. (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 6128 . 2 Rel ( 𝑥𝐶 𝐴𝐵)
2 reliun 5828 . . 3 (Rel 𝑥𝐶 (𝐴𝐵) ↔ ∀𝑥𝐶 Rel (𝐴𝐵))
3 relco 6128 . . . 4 Rel (𝐴𝐵)
43a1i 11 . . 3 (𝑥𝐶 → Rel (𝐴𝐵))
52, 4mprgbir 3065 . 2 Rel 𝑥𝐶 (𝐴𝐵)
6 eliun 4999 . . . . . . . 8 (⟨𝑤, 𝑧⟩ ∈ 𝑥𝐶 𝐴 ↔ ∃𝑥𝐶𝑤, 𝑧⟩ ∈ 𝐴)
7 df-br 5148 . . . . . . . 8 (𝑤 𝑥𝐶 𝐴𝑧 ↔ ⟨𝑤, 𝑧⟩ ∈ 𝑥𝐶 𝐴)
8 df-br 5148 . . . . . . . . 9 (𝑤𝐴𝑧 ↔ ⟨𝑤, 𝑧⟩ ∈ 𝐴)
98rexbii 3091 . . . . . . . 8 (∃𝑥𝐶 𝑤𝐴𝑧 ↔ ∃𝑥𝐶𝑤, 𝑧⟩ ∈ 𝐴)
106, 7, 93bitr4i 303 . . . . . . 7 (𝑤 𝑥𝐶 𝐴𝑧 ↔ ∃𝑥𝐶 𝑤𝐴𝑧)
1110anbi2i 623 . . . . . 6 ((𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧) ↔ (𝑦𝐵𝑤 ∧ ∃𝑥𝐶 𝑤𝐴𝑧))
12 r19.42v 3188 . . . . . 6 (∃𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧) ↔ (𝑦𝐵𝑤 ∧ ∃𝑥𝐶 𝑤𝐴𝑧))
1311, 12bitr4i 278 . . . . 5 ((𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧) ↔ ∃𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
1413exbii 1844 . . . 4 (∃𝑤(𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧) ↔ ∃𝑤𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
15 rexcom4 3285 . . . 4 (∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧) ↔ ∃𝑤𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
1614, 15bitr4i 278 . . 3 (∃𝑤(𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
17 vex 3481 . . . 4 𝑦 ∈ V
18 vex 3481 . . . 4 𝑧 ∈ V
1917, 18opelco 5884 . . 3 (⟨𝑦, 𝑧⟩ ∈ ( 𝑥𝐶 𝐴𝐵) ↔ ∃𝑤(𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧))
20 eliun 4999 . . . 4 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵) ↔ ∃𝑥𝐶𝑦, 𝑧⟩ ∈ (𝐴𝐵))
2117, 18opelco 5884 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2221rexbii 3091 . . . 4 (∃𝑥𝐶𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2320, 22bitri 275 . . 3 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2416, 19, 233bitr4i 303 . 2 (⟨𝑦, 𝑧⟩ ∈ ( 𝑥𝐶 𝐴𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵))
251, 5, 24eqrelriiv 5802 1 ( 𝑥𝐶 𝐴𝐵) = 𝑥𝐶 (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1536  wex 1775  wcel 2105  wrex 3067  cop 4636   ciun 4995   class class class wbr 5147  ccom 5692  Rel wrel 5693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-iun 4997  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-co 5697
This theorem is referenced by:  trclfvcom  43712  trclfvdecomr  43717  cotrclrcl  43731
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