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Theorem cnviun 40725
 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 5940 . 2 Rel 𝑥𝐴 𝐵
2 reliun 5659 . . 3 (Rel 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 Rel 𝐵)
3 relcnv 5940 . . . 4 Rel 𝐵
43a1i 11 . . 3 (𝑥𝐴 → Rel 𝐵)
52, 4mprgbir 3086 . 2 Rel 𝑥𝐴 𝐵
6 vex 3414 . . . . . 6 𝑦 ∈ V
7 vex 3414 . . . . . 6 𝑧 ∈ V
86, 7opelcnv 5722 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ 𝐵 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐵)
98bicomi 227 . . . 4 (⟨𝑧, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵)
109rexbii 3176 . . 3 (∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
116, 7opelcnv 5722 . . . 4 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵)
12 eliun 4888 . . . 4 (⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵)
1311, 12bitri 278 . . 3 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵)
14 eliun 4888 . . 3 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
1510, 13, 143bitr4i 307 . 2 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵)
161, 5, 15eqrelriiv 5633 1 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵
 Colors of variables: wff setvar class Syntax hints:   = wceq 1539   ∈ wcel 2112  ∃wrex 3072  ⟨cop 4529  ∪ ciun 4884  ◡ccnv 5524  Rel wrel 5530 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-iun 4886  df-br 5034  df-opab 5096  df-xp 5531  df-rel 5532  df-cnv 5533 This theorem is referenced by:  cnvtrclfv  40799
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