Theorem cnviun 37942

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 5503	. 2 ⊢ Rel ◡∪ 𝑥 ∈ 𝐴 𝐵
2		reliun 5239	. . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 Rel ◡𝐵)
3		relcnv 5503	. . . 4 ⊢ Rel ◡𝐵
4	3	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → Rel ◡𝐵)
5	2, 4	mprgbir 2927	. 2 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ◡𝐵
6		vex 3203	. . . . . 6 ⊢ 𝑦 ∈ V
7		vex 3203	. . . . . 6 ⊢ 𝑧 ∈ V
8	6, 7	opelcnv 5304	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ ◡𝐵 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐵)
9	8	bicomi 214	. . . 4 ⊢ (⟨𝑧, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ ◡𝐵)
10	9	rexbii 3041	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ⟨𝑧, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ ◡𝐵)
11	6, 7	opelcnv 5304	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ ◡∪ 𝑥 ∈ 𝐴 𝐵 ↔ ⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
12		eliun 4524	. . . 4 ⊢ (⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑧, 𝑦⟩ ∈ 𝐵)
13	11, 12	bitri 264	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ ◡∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑧, 𝑦⟩ ∈ 𝐵)
14		eliun 4524	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ ◡𝐵)
15	10, 13, 14	3bitr4i 292	. 2 ⊢ (⟨𝑦, 𝑧⟩ ∈ ◡∪ 𝑥 ∈ 𝐴 𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 ◡𝐵)
16	1, 5, 15	eqrelriiv 5214	1 ⊢ ◡∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ◡𝐵

Syntax hints:   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  ⟨cop 4183  ∪ ciun 4520  ^◡ccnv 5113  Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-iun 4522  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122

This theorem is referenced by:  cnvtrclfv  38016