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Theorem coiun1 37422
Description: Composition with an indexed union. Proof analgous to that of coiun 5604. (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 5592 . 2 Rel ( 𝑥𝐶 𝐴𝐵)
2 reliun 5200 . . 3 (Rel 𝑥𝐶 (𝐴𝐵) ↔ ∀𝑥𝐶 Rel (𝐴𝐵))
3 relco 5592 . . . 4 Rel (𝐴𝐵)
43a1i 11 . . 3 (𝑥𝐶 → Rel (𝐴𝐵))
52, 4mprgbir 2922 . 2 Rel 𝑥𝐶 (𝐴𝐵)
6 eliun 4490 . . . . . . . 8 (⟨𝑤, 𝑧⟩ ∈ 𝑥𝐶 𝐴 ↔ ∃𝑥𝐶𝑤, 𝑧⟩ ∈ 𝐴)
7 df-br 4614 . . . . . . . 8 (𝑤 𝑥𝐶 𝐴𝑧 ↔ ⟨𝑤, 𝑧⟩ ∈ 𝑥𝐶 𝐴)
8 df-br 4614 . . . . . . . . 9 (𝑤𝐴𝑧 ↔ ⟨𝑤, 𝑧⟩ ∈ 𝐴)
98rexbii 3034 . . . . . . . 8 (∃𝑥𝐶 𝑤𝐴𝑧 ↔ ∃𝑥𝐶𝑤, 𝑧⟩ ∈ 𝐴)
106, 7, 93bitr4i 292 . . . . . . 7 (𝑤 𝑥𝐶 𝐴𝑧 ↔ ∃𝑥𝐶 𝑤𝐴𝑧)
1110anbi2i 729 . . . . . 6 ((𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧) ↔ (𝑦𝐵𝑤 ∧ ∃𝑥𝐶 𝑤𝐴𝑧))
12 r19.42v 3084 . . . . . 6 (∃𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧) ↔ (𝑦𝐵𝑤 ∧ ∃𝑥𝐶 𝑤𝐴𝑧))
1311, 12bitr4i 267 . . . . 5 ((𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧) ↔ ∃𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
1413exbii 1771 . . . 4 (∃𝑤(𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧) ↔ ∃𝑤𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
15 rexcom4 3211 . . . 4 (∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧) ↔ ∃𝑤𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
1614, 15bitr4i 267 . . 3 (∃𝑤(𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
17 vex 3189 . . . 4 𝑦 ∈ V
18 vex 3189 . . . 4 𝑧 ∈ V
1917, 18opelco 5253 . . 3 (⟨𝑦, 𝑧⟩ ∈ ( 𝑥𝐶 𝐴𝐵) ↔ ∃𝑤(𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧))
20 eliun 4490 . . . 4 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵) ↔ ∃𝑥𝐶𝑦, 𝑧⟩ ∈ (𝐴𝐵))
2117, 18opelco 5253 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2221rexbii 3034 . . . 4 (∃𝑥𝐶𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2320, 22bitri 264 . . 3 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2416, 19, 233bitr4i 292 . 2 (⟨𝑦, 𝑧⟩ ∈ ( 𝑥𝐶 𝐴𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵))
251, 5, 24eqrelriiv 5175 1 ( 𝑥𝐶 𝐴𝐵) = 𝑥𝐶 (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wex 1701  wcel 1987  wrex 2908  cop 4154   ciun 4485   class class class wbr 4613  ccom 5078  Rel wrel 5079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-iun 4487  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-co 5083
This theorem is referenced by:  trclfvcom  37493  trclfvdecomr  37498  cotrclrcl  37512
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