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Theorem cnviun 37420
 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 5462 . 2 Rel 𝑥𝐴 𝐵
2 reliun 5200 . . 3 (Rel 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 Rel 𝐵)
3 relcnv 5462 . . . 4 Rel 𝐵
43a1i 11 . . 3 (𝑥𝐴 → Rel 𝐵)
52, 4mprgbir 2922 . 2 Rel 𝑥𝐴 𝐵
6 vex 3189 . . . . . 6 𝑦 ∈ V
7 vex 3189 . . . . . 6 𝑧 ∈ V
86, 7opelcnv 5264 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ 𝐵 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐵)
98bicomi 214 . . . 4 (⟨𝑧, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵)
109rexbii 3034 . . 3 (∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
116, 7opelcnv 5264 . . . 4 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵)
12 eliun 4490 . . . 4 (⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵)
1311, 12bitri 264 . . 3 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵)
14 eliun 4490 . . 3 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
1510, 13, 143bitr4i 292 . 2 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵)
161, 5, 15eqrelriiv 5175 1 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵
 Colors of variables: wff setvar class Syntax hints:   = wceq 1480   ∈ wcel 1987  ∃wrex 2908  ⟨cop 4154  ∪ ciun 4485  ◡ccnv 5073  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-cnv 5082 This theorem is referenced by:  cnvtrclfv  37494
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