Theorem cnviun 36961

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 5422	. 2 ⊢ Rel ◡∪ 𝑥 ∈ 𝐴 𝐵
2		reliun 5162	. . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 Rel ◡𝐵)
3		relcnv 5422	. . . 4 ⊢ Rel ◡𝐵
4	3	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → Rel ◡𝐵)
5	2, 4	mprgbir 2911	. 2 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ◡𝐵
6		vex 3176	. . . . . 6 ⊢ 𝑦 ∈ V
7		vex 3176	. . . . . 6 ⊢ 𝑧 ∈ V
8	6, 7	opelcnv 5226	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ ◡𝐵 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐵)
9	8	bicomi 213	. . . 4 ⊢ (⟨𝑧, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ ◡𝐵)
10	9	rexbii 3023	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ⟨𝑧, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ ◡𝐵)
11	6, 7	opelcnv 5226	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ ◡∪ 𝑥 ∈ 𝐴 𝐵 ↔ ⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
12		eliun 4460	. . . 4 ⊢ (⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑧, 𝑦⟩ ∈ 𝐵)
13	11, 12	bitri 263	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ ◡∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑧, 𝑦⟩ ∈ 𝐵)
14		eliun 4460	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ ◡𝐵)
15	10, 13, 14	3bitr4i 291	. 2 ⊢ (⟨𝑦, 𝑧⟩ ∈ ◡∪ 𝑥 ∈ 𝐴 𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 ◡𝐵)
16	1, 5, 15	eqrelriiv 5137	1 ⊢ ◡∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ◡𝐵

Syntax hints:   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  ⟨cop 4131  ∪ ciun 4455  ^◡ccnv 5037  Rel wrel 5043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-iun 4457  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046

This theorem is referenced by:  cnvtrclfv  37035