Theorem cnviun 43621

Description: Converse of indexed union. (Contributed by RP, 20-Jun-2020.)

Assertion

Ref	Expression
cnviun	⊢ ◡∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ◡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.

Step	Hyp	Ref	Expression
1		relcnv 6091	. 2 ⊢ Rel ◡∪ 𝑥 ∈ 𝐴 𝐵
2		reliun 5795	. . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 Rel ◡𝐵)
3		relcnv 6091	. . . 4 ⊢ Rel ◡𝐵
4	3	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → Rel ◡𝐵)
5	2, 4	mprgbir 3058	. 2 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ◡𝐵
6		vex 3463	. . . . . 6 ⊢ 𝑦 ∈ V
7		vex 3463	. . . . . 6 ⊢ 𝑧 ∈ V
8	6, 7	opelcnv 5861	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ ◡𝐵 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐵)
9	8	bicomi 224	. . . 4 ⊢ (⟨𝑧, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ ◡𝐵)
10	9	rexbii 3083	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ⟨𝑧, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ ◡𝐵)
11	6, 7	opelcnv 5861	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ ◡∪ 𝑥 ∈ 𝐴 𝐵 ↔ ⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
12		eliun 4971	. . . 4 ⊢ (⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑧, 𝑦⟩ ∈ 𝐵)
13	11, 12	bitri 275	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ ◡∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑧, 𝑦⟩ ∈ 𝐵)
14		eliun 4971	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ ◡𝐵)
15	10, 13, 14	3bitr4i 303	. 2 ⊢ (⟨𝑦, 𝑧⟩ ∈ ◡∪ 𝑥 ∈ 𝐴 𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 ◡𝐵)
16	1, 5, 15	eqrelriiv 5769	1 ⊢ ◡∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ◡𝐵

Syntax hints:   = wceq 1540   ∈ wcel 2108  ∃wrex 3060  ⟨cop 4607  ∪ ciun 4967  ^◡ccnv 5653  Rel wrel 5659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-iun 4969  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-cnv 5662

This theorem is referenced by:  cnvtrclfv  43695