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Theorem coiun1 40348
Description: Composition with an indexed union. Proof analgous to that of coiun 6076. (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 6064 . 2 Rel ( 𝑥𝐶 𝐴𝐵)
2 reliun 5653 . . 3 (Rel 𝑥𝐶 (𝐴𝐵) ↔ ∀𝑥𝐶 Rel (𝐴𝐵))
3 relco 6064 . . . 4 Rel (𝐴𝐵)
43a1i 11 . . 3 (𝑥𝐶 → Rel (𝐴𝐵))
52, 4mprgbir 3121 . 2 Rel 𝑥𝐶 (𝐴𝐵)
6 eliun 4885 . . . . . . . 8 (⟨𝑤, 𝑧⟩ ∈ 𝑥𝐶 𝐴 ↔ ∃𝑥𝐶𝑤, 𝑧⟩ ∈ 𝐴)
7 df-br 5031 . . . . . . . 8 (𝑤 𝑥𝐶 𝐴𝑧 ↔ ⟨𝑤, 𝑧⟩ ∈ 𝑥𝐶 𝐴)
8 df-br 5031 . . . . . . . . 9 (𝑤𝐴𝑧 ↔ ⟨𝑤, 𝑧⟩ ∈ 𝐴)
98rexbii 3210 . . . . . . . 8 (∃𝑥𝐶 𝑤𝐴𝑧 ↔ ∃𝑥𝐶𝑤, 𝑧⟩ ∈ 𝐴)
106, 7, 93bitr4i 306 . . . . . . 7 (𝑤 𝑥𝐶 𝐴𝑧 ↔ ∃𝑥𝐶 𝑤𝐴𝑧)
1110anbi2i 625 . . . . . 6 ((𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧) ↔ (𝑦𝐵𝑤 ∧ ∃𝑥𝐶 𝑤𝐴𝑧))
12 r19.42v 3303 . . . . . 6 (∃𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧) ↔ (𝑦𝐵𝑤 ∧ ∃𝑥𝐶 𝑤𝐴𝑧))
1311, 12bitr4i 281 . . . . 5 ((𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧) ↔ ∃𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
1413exbii 1849 . . . 4 (∃𝑤(𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧) ↔ ∃𝑤𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
15 rexcom4 3212 . . . 4 (∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧) ↔ ∃𝑤𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
1614, 15bitr4i 281 . . 3 (∃𝑤(𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
17 vex 3444 . . . 4 𝑦 ∈ V
18 vex 3444 . . . 4 𝑧 ∈ V
1917, 18opelco 5706 . . 3 (⟨𝑦, 𝑧⟩ ∈ ( 𝑥𝐶 𝐴𝐵) ↔ ∃𝑤(𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧))
20 eliun 4885 . . . 4 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵) ↔ ∃𝑥𝐶𝑦, 𝑧⟩ ∈ (𝐴𝐵))
2117, 18opelco 5706 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2221rexbii 3210 . . . 4 (∃𝑥𝐶𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2320, 22bitri 278 . . 3 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2416, 19, 233bitr4i 306 . 2 (⟨𝑦, 𝑧⟩ ∈ ( 𝑥𝐶 𝐴𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵))
251, 5, 24eqrelriiv 5627 1 ( 𝑥𝐶 𝐴𝐵) = 𝑥𝐶 (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wex 1781  wcel 2111  wrex 3107  cop 4531   ciun 4881   class class class wbr 5030  ccom 5523  Rel wrel 5524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-iun 4883  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-co 5528
This theorem is referenced by:  trclfvcom  40419  trclfvdecomr  40424  cotrclrcl  40438
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