Theorem cnviun 44649

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 6100	. 2 ⊢ Rel ◡∪ 𝑥 ∈ 𝐴 𝐵
2		reliun 5794	. . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 Rel ◡𝐵)
3		relcnv 6100	. . . 4 ⊢ Rel ◡𝐵
4	3	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → Rel ◡𝐵)
5	2, 4	mprgbir 3084	. 2 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ◡𝐵
6		vex 3455	. . . . . 6 ⊢ 𝑦 ∈ V
7		vex 3455	. . . . . 6 ⊢ 𝑧 ∈ V
8	6, 7	opelcnv 5859	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ ◡𝐵 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐵)
9	8	bicomi 227	. . . 4 ⊢ (⟨𝑧, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ ◡𝐵)
10	9	rexbii 3110	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ⟨𝑧, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ ◡𝐵)
11	6, 7	opelcnv 5859	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ ◡∪ 𝑥 ∈ 𝐴 𝐵 ↔ ⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
12		eliun 4955	. . . 4 ⊢ (⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑧, 𝑦⟩ ∈ 𝐵)
13	11, 12	bitri 278	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ ◡∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑧, 𝑦⟩ ∈ 𝐵)
14		eliun 4955	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ ◡𝐵)
15	10, 13, 14	3bitr4i 306	. 2 ⊢ (⟨𝑦, 𝑧⟩ ∈ ◡∪ 𝑥 ∈ 𝐴 𝐵 ↔ ⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 ◡𝐵)
16	1, 5, 15	eqrelriiv 5766	1 ⊢ ◡∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ◡𝐵

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  ∃wrex 3087  ⟨cop 4590  ∪ ciun 4951  ^◡ccnv 5650  Rel wrel 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-iun 4953  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-cnv 5659

This theorem is used by:  cnvtrclfv  44723