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Theorem coiun1 38470
Description: Composition with an indexed union. Proof analgous to that of coiun 5789. (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 5777 . 2 Rel ( 𝑥𝐶 𝐴𝐵)
2 reliun 5378 . . 3 (Rel 𝑥𝐶 (𝐴𝐵) ↔ ∀𝑥𝐶 Rel (𝐴𝐵))
3 relco 5777 . . . 4 Rel (𝐴𝐵)
43a1i 11 . . 3 (𝑥𝐶 → Rel (𝐴𝐵))
52, 4mprgbir 3076 . 2 Rel 𝑥𝐶 (𝐴𝐵)
6 eliun 4658 . . . . . . . 8 (⟨𝑤, 𝑧⟩ ∈ 𝑥𝐶 𝐴 ↔ ∃𝑥𝐶𝑤, 𝑧⟩ ∈ 𝐴)
7 df-br 4787 . . . . . . . 8 (𝑤 𝑥𝐶 𝐴𝑧 ↔ ⟨𝑤, 𝑧⟩ ∈ 𝑥𝐶 𝐴)
8 df-br 4787 . . . . . . . . 9 (𝑤𝐴𝑧 ↔ ⟨𝑤, 𝑧⟩ ∈ 𝐴)
98rexbii 3189 . . . . . . . 8 (∃𝑥𝐶 𝑤𝐴𝑧 ↔ ∃𝑥𝐶𝑤, 𝑧⟩ ∈ 𝐴)
106, 7, 93bitr4i 292 . . . . . . 7 (𝑤 𝑥𝐶 𝐴𝑧 ↔ ∃𝑥𝐶 𝑤𝐴𝑧)
1110anbi2i 609 . . . . . 6 ((𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧) ↔ (𝑦𝐵𝑤 ∧ ∃𝑥𝐶 𝑤𝐴𝑧))
12 r19.42v 3240 . . . . . 6 (∃𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧) ↔ (𝑦𝐵𝑤 ∧ ∃𝑥𝐶 𝑤𝐴𝑧))
1311, 12bitr4i 267 . . . . 5 ((𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧) ↔ ∃𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
1413exbii 1924 . . . 4 (∃𝑤(𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧) ↔ ∃𝑤𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
15 rexcom4 3377 . . . 4 (∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧) ↔ ∃𝑤𝑥𝐶 (𝑦𝐵𝑤𝑤𝐴𝑧))
1614, 15bitr4i 267 . . 3 (∃𝑤(𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧) ↔ ∃𝑥𝐶𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
17 vex 3354 . . . 4 𝑦 ∈ V
18 vex 3354 . . . 4 𝑧 ∈ V
1917, 18opelco 5432 . . 3 (⟨𝑦, 𝑧⟩ ∈ ( 𝑥𝐶 𝐴𝐵) ↔ ∃𝑤(𝑦𝐵𝑤𝑤 𝑥𝐶 𝐴𝑧))
20 eliun 4658 . . . 4 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐶 (𝐴𝐵) ↔ ∃𝑥𝐶𝑦, 𝑧⟩ ∈ (𝐴𝐵))
2117, 18opelco 5432 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧))
2221rexbii 3189 . . . 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 5354 1 ( 𝑥𝐶 𝐴𝐵) = 𝑥𝐶 (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1631  wex 1852  wcel 2145  wrex 3062  cop 4322   ciun 4654   class class class wbr 4786  ccom 5253  Rel wrel 5254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-iun 4656  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-co 5258
This theorem is referenced by:  trclfvcom  38541  trclfvdecomr  38546  cotrclrcl  38560
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