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Theorem cnviun 43639
Description: Converse of indexed union. (Contributed by RP, 20-Jun-2020.)
Assertion
Ref Expression
cnviun 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem cnviun
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 6124 . 2 Rel 𝑥𝐴 𝐵
2 reliun 5828 . . 3 (Rel 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 Rel 𝐵)
3 relcnv 6124 . . . 4 Rel 𝐵
43a1i 11 . . 3 (𝑥𝐴 → Rel 𝐵)
52, 4mprgbir 3065 . 2 Rel 𝑥𝐴 𝐵
6 vex 3481 . . . . . 6 𝑦 ∈ V
7 vex 3481 . . . . . 6 𝑧 ∈ V
86, 7opelcnv 5894 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ 𝐵 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐵)
98bicomi 224 . . . 4 (⟨𝑧, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵)
109rexbii 3091 . . 3 (∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
116, 7opelcnv 5894 . . . 4 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵)
12 eliun 4999 . . . 4 (⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵)
1311, 12bitri 275 . . 3 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵)
14 eliun 4999 . . 3 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
1510, 13, 143bitr4i 303 . 2 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵)
161, 5, 15eqrelriiv 5802 1 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1536  wcel 2105  wrex 3067  cop 4636   ciun 4995  ccnv 5687  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-cnv 5696
This theorem is referenced by:  cnvtrclfv  43713
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