iunrelexp0 - Mathbox for Richard Penner

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Theorem iunrelexp0 39553

Description: Simplification of zeroth power of indexed union of powers of relations. (Contributed by RP, 19-Jun-2020.)

**Assertion**
Ref	Expression
iunrelexp0	⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀ ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → (∪ 𝑥 ∈ 𝑍 (𝑅↑_𝑟𝑥)↑_𝑟0) = (𝑅↑_𝑟0))

Distinct variable groups: 𝑥,𝑅 𝑥,𝑉 𝑥,𝑍

**Proof of Theorem iunrelexp0**
Step	Hyp	Ref	Expression
1		df-pr 4481	. . . . . . 7 ⊢ {0, 1} = ({0} ∪ {1})
2	1	ineq1i 4111	. . . . . 6 ⊢ ({0, 1} ∩ 𝑍) = (({0} ∪ {1}) ∩ 𝑍)
3		indir 4178	. . . . . 6 ⊢ (({0} ∪ {1}) ∩ 𝑍) = (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))
4	2, 3	eqtr2i 2822	. . . . 5 ⊢ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) = ({0, 1} ∩ 𝑍)
5	4	uneq1i 4062	. . . 4 ⊢ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) = (({0, 1} ∩ 𝑍) ∪ 𝑍)
6		inss2 4132	. . . . 5 ⊢ ({0, 1} ∩ 𝑍) ⊆ 𝑍
7		ssequn1 4083	. . . . 5 ⊢ (({0, 1} ∩ 𝑍) ⊆ 𝑍 ↔ (({0, 1} ∩ 𝑍) ∪ 𝑍) = 𝑍)
8	6, 7	mpbi 231	. . . 4 ⊢ (({0, 1} ∩ 𝑍) ∪ 𝑍) = 𝑍
9	5, 8	eqtr2i 2822	. . 3 ⊢ 𝑍 = ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)
10		iuneq1 4846	. . . 4 ⊢ (𝑍 = ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) → ∪ 𝑥 ∈ 𝑍 (𝑅↑_𝑟𝑥) = ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥))
11	10	oveq1d 7038	. . 3 ⊢ (𝑍 = ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) → (∪ 𝑥 ∈ 𝑍 (𝑅↑_𝑟𝑥)↑_𝑟0) = (∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥)↑_𝑟0))
12	9, 11	ax-mp 5	. 2 ⊢ (∪ 𝑥 ∈ 𝑍 (𝑅↑_𝑟𝑥)↑_𝑟0) = (∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥)↑_𝑟0)
13		dmiun 5675	. . . . . . 7 ⊢ dom ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) = ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)dom (𝑅↑_𝑟𝑥)
14		iunxun 4921	. . . . . . 7 ⊢ ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)dom (𝑅↑_𝑟𝑥) = (∪ 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥))
15		iunxun 4921	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅↑_𝑟𝑥) = (∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥))
16	15	equncomi 4058	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅↑_𝑟𝑥) = (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥))
17	16	uneq1i 4062	. . . . . . . 8 ⊢ (∪ 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥)) = ((∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥)) ∪ ∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥))
18	17	equncomi 4058	. . . . . . 7 ⊢ (∪ 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥)) = (∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥)))
19	13, 14, 18	3eqtri 2825	. . . . . 6 ⊢ dom ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) = (∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥)))
20		rniun 5889	. . . . . . 7 ⊢ ran ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) = ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)ran (𝑅↑_𝑟𝑥)
21		iunxun 4921	. . . . . . 7 ⊢ ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)ran (𝑅↑_𝑟𝑥) = (∪ 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))ran (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥))
22		iunxun 4921	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))ran (𝑅↑_𝑟𝑥) = (∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))
23	22	uneq1i 4062	. . . . . . 7 ⊢ (∪ 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))ran (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥)) = ((∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥))
24	20, 21, 23	3eqtri 2825	. . . . . 6 ⊢ ran ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) = ((∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥))
25	19, 24	uneq12i 4064	. . . . 5 ⊢ (dom ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) ∪ ran ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥)) = ((∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥))) ∪ ((∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥)))
26		uncom 4056	. . . . . . 7 ⊢ (∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥))) = ((∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥)) ∪ ∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥))
27	26	uneq1i 4062	. . . . . 6 ⊢ ((∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥))) ∪ ((∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥))) = (((∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥)) ∪ ∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥)) ∪ ((∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥)))
28		un4 4072	. . . . . 6 ⊢ (((∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥)) ∪ ∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥)) ∪ ((∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥))) = (((∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))) ∪ (∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥)))
29	27, 28	eqtri 2821	. . . . 5 ⊢ ((∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥))) ∪ ((∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥))) = (((∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))) ∪ (∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥)))
30		uncom 4056	. . . . . . . 8 ⊢ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥)) = (∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥))
31	30	uneq1i 4062	. . . . . . 7 ⊢ ((∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))) = ((∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)))
32		un4 4072	. . . . . . 7 ⊢ ((∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))) = ((∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)))
33	31, 32	eqtri 2821	. . . . . 6 ⊢ ((∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))) = ((∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)))
34	33	uneq1i 4062	. . . . 5 ⊢ (((∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))) ∪ (∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥))) = (((∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))) ∪ (∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥)))
35	25, 29, 34	3eqtri 2825	. . . 4 ⊢ (dom ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) ∪ ran ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥)) = (((∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))) ∪ (∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥)))
36		df-ne 2987	. . . . . . . . . 10 ⊢ (({0, 1} ∩ 𝑍) ≠ ∅ ↔ ¬ ({0, 1} ∩ 𝑍) = ∅)
37		incom 4105	. . . . . . . . . . . . . . 15 ⊢ ({0, 1} ∩ 𝑍) = (𝑍 ∩ {0, 1})
38	1	ineq2i 4112	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∩ {0, 1}) = (𝑍 ∩ ({0} ∪ {1}))
39		indi 4176	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∩ ({0} ∪ {1})) = ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1}))
40	37, 38, 39	3eqtri 2825	. . . . . . . . . . . . . 14 ⊢ ({0, 1} ∩ 𝑍) = ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1}))
41	40	eqeq1i 2802	. . . . . . . . . . . . 13 ⊢ (({0, 1} ∩ 𝑍) = ∅ ↔ ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1})) = ∅)
42		un00 4314	. . . . . . . . . . . . 13 ⊢ (((𝑍 ∩ {0}) = ∅ ∧ (𝑍 ∩ {1}) = ∅) ↔ ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1})) = ∅)
43		anor 977	. . . . . . . . . . . . 13 ⊢ (((𝑍 ∩ {0}) = ∅ ∧ (𝑍 ∩ {1}) = ∅) ↔ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
44	41, 42, 43	3bitr2i 300	. . . . . . . . . . . 12 ⊢ (({0, 1} ∩ 𝑍) = ∅ ↔ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
45	44	notbii 321	. . . . . . . . . . 11 ⊢ (¬ ({0, 1} ∩ 𝑍) = ∅ ↔ ¬ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
46		notnotb 316	. . . . . . . . . . 11 ⊢ ((¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅) ↔ ¬ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
47		disjsn 4560	. . . . . . . . . . . . . 14 ⊢ ((𝑍 ∩ {0}) = ∅ ↔ ¬ 0 ∈ 𝑍)
48	47	notbii 321	. . . . . . . . . . . . 13 ⊢ (¬ (𝑍 ∩ {0}) = ∅ ↔ ¬ ¬ 0 ∈ 𝑍)
49		notnotb 316	. . . . . . . . . . . . 13 ⊢ (0 ∈ 𝑍 ↔ ¬ ¬ 0 ∈ 𝑍)
50	48, 49	bitr4i 279	. . . . . . . . . . . 12 ⊢ (¬ (𝑍 ∩ {0}) = ∅ ↔ 0 ∈ 𝑍)
51		disjsn 4560	. . . . . . . . . . . . . 14 ⊢ ((𝑍 ∩ {1}) = ∅ ↔ ¬ 1 ∈ 𝑍)
52	51	notbii 321	. . . . . . . . . . . . 13 ⊢ (¬ (𝑍 ∩ {1}) = ∅ ↔ ¬ ¬ 1 ∈ 𝑍)
53		notnotb 316	. . . . . . . . . . . . 13 ⊢ (1 ∈ 𝑍 ↔ ¬ ¬ 1 ∈ 𝑍)
54	52, 53	bitr4i 279	. . . . . . . . . . . 12 ⊢ (¬ (𝑍 ∩ {1}) = ∅ ↔ 1 ∈ 𝑍)
55	50, 54	orbi12i 909	. . . . . . . . . . 11 ⊢ ((¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅) ↔ (0 ∈ 𝑍 ∨ 1 ∈ 𝑍))
56	45, 46, 55	3bitr2i 300	. . . . . . . . . 10 ⊢ (¬ ({0, 1} ∩ 𝑍) = ∅ ↔ (0 ∈ 𝑍 ∨ 1 ∈ 𝑍))
57	36, 56	sylbb 220	. . . . . . . . 9 ⊢ (({0, 1} ∩ 𝑍) ≠ ∅ → (0 ∈ 𝑍 ∨ 1 ∈ 𝑍))
58		simpl 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → 0 ∈ 𝑍)
59	58	snssd 4655	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → {0} ⊆ 𝑍)
60		df-ss 3880	. . . . . . . . . . . . . . . . . . 19 ⊢ ({0} ⊆ 𝑍 ↔ ({0} ∩ 𝑍) = {0})
61	59, 60	sylib 219	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ({0} ∩ 𝑍) = {0})
62	61	iuneq1d 4857	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) = ∪ 𝑥 ∈ {0}dom (𝑅↑_𝑟𝑥))
63		c0ex 10488	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
64		oveq2 7031	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 0 → (𝑅↑_𝑟𝑥) = (𝑅↑_𝑟0))
65	64	dmeqd 5667	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 0 → dom (𝑅↑_𝑟𝑥) = dom (𝑅↑_𝑟0))
66	63, 65	iunxsn 4918	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑥 ∈ {0}dom (𝑅↑_𝑟𝑥) = dom (𝑅↑_𝑟0)
67	62, 66	syl6eq 2849	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) = dom (𝑅↑_𝑟0))
68		relexp0g 14219	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑_𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
69	68	ad2antll 725	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (𝑅↑_𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
70	69	dmeqd 5667	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → dom (𝑅↑_𝑟0) = dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
71		dmresi 5806	. . . . . . . . . . . . . . . . 17 ⊢ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
72	70, 71	syl6eq 2849	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → dom (𝑅↑_𝑟0) = (dom 𝑅 ∪ ran 𝑅))
73	67, 72	eqtrd 2833	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) = (dom 𝑅 ∪ ran 𝑅))
74	61	iuneq1d 4857	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) = ∪ 𝑥 ∈ {0}ran (𝑅↑_𝑟𝑥))
75	64	rneqd 5697	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 0 → ran (𝑅↑_𝑟𝑥) = ran (𝑅↑_𝑟0))
76	63, 75	iunxsn 4918	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑥 ∈ {0}ran (𝑅↑_𝑟𝑥) = ran (𝑅↑_𝑟0)
77	74, 76	syl6eq 2849	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) = ran (𝑅↑_𝑟0))
78	69	rneqd 5697	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ran (𝑅↑_𝑟0) = ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
79		rnresi 5826	. . . . . . . . . . . . . . . . 17 ⊢ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
80	78, 79	syl6eq 2849	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ran (𝑅↑_𝑟0) = (dom 𝑅 ∪ ran 𝑅))
81	77, 80	eqtrd 2833	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) = (dom 𝑅 ∪ ran 𝑅))
82	73, 81	uneq12d 4067	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) = ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅)))
83		unidm 4055	. . . . . . . . . . . . . 14 ⊢ ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
84	82, 83	syl6eq 2849	. . . . . . . . . . . . 13 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) = (dom 𝑅 ∪ ran 𝑅))
85	84	uneq1d 4065	. . . . . . . . . . . 12 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ((∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))) = ((dom 𝑅 ∪ ran 𝑅) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))))
86		relexpdmg 14239	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
87	86	expcom 414	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ ℕ₀ → dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
88	87	ralrimiv 3150	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ 𝑉 → ∀𝑥 ∈ ℕ₀ dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
89	88	ad2antll 725	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∀𝑥 ∈ ℕ₀ dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
90		olc 863	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑍 ⊆ ℕ₀ → ({1} ⊆ ℕ₀ ∨ 𝑍 ⊆ ℕ₀))
91	90	ad2antrl 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ({1} ⊆ ℕ₀ ∨ 𝑍 ⊆ ℕ₀))
92		inss 4141	. . . . . . . . . . . . . . . . . . . 20 ⊢ (({1} ⊆ ℕ₀ ∨ 𝑍 ⊆ ℕ₀) → ({1} ∩ 𝑍) ⊆ ℕ₀)
93	91, 92	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ({1} ∩ 𝑍) ⊆ ℕ₀)
94	93	sseld 3894	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (𝑥 ∈ ({1} ∩ 𝑍) → 𝑥 ∈ ℕ₀))
95	94	imim1d 82	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ((𝑥 ∈ ℕ₀ → dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)) → (𝑥 ∈ ({1} ∩ 𝑍) → dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))))
96	95	ralimdv2 3145	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (∀𝑥 ∈ ℕ₀ dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) → ∀𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
97	89, 96	mpd 15	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∀𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
98		iunss 4874	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
99	97, 98	sylibr 235	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
100		relexprng 14243	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
101	100	expcom 414	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ ℕ₀ → ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
102	101	ralrimiv 3150	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ 𝑉 → ∀𝑥 ∈ ℕ₀ ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
103	102	ad2antll 725	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∀𝑥 ∈ ℕ₀ ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
104	94	imim1d 82	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ((𝑥 ∈ ℕ₀ → ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)) → (𝑥 ∈ ({1} ∩ 𝑍) → ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))))
105	104	ralimdv2 3145	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (∀𝑥 ∈ ℕ₀ ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) → ∀𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
106	103, 105	mpd 15	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∀𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
107		iunss 4874	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
108	106, 107	sylibr 235	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
109	99, 108	unssd 4089	. . . . . . . . . . . . 13 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
110		ssequn2 4086	. . . . . . . . . . . . 13 ⊢ ((∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ((dom 𝑅 ∪ ran 𝑅) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
111	109, 110	sylib 219	. . . . . . . . . . . 12 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ((dom 𝑅 ∪ ran 𝑅) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
112	85, 111	eqtrd 2833	. . . . . . . . . . 11 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ((∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
113	112	ex 413	. . . . . . . . . 10 ⊢ (0 ∈ 𝑍 → ((𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉) → ((∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅)))
114		simpl 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → 1 ∈ 𝑍)
115	114	snssd 4655	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → {1} ⊆ 𝑍)
116		df-ss 3880	. . . . . . . . . . . . . . . . . 18 ⊢ ({1} ⊆ 𝑍 ↔ ({1} ∩ 𝑍) = {1})
117	115, 116	sylib 219	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ({1} ∩ 𝑍) = {1})
118	117	iuneq1d 4857	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) = ∪ 𝑥 ∈ {1}dom (𝑅↑_𝑟𝑥))
119		1ex 10490	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ V
120		oveq2 7031	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 1 → (𝑅↑_𝑟𝑥) = (𝑅↑_𝑟1))
121	120	dmeqd 5667	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 1 → dom (𝑅↑_𝑟𝑥) = dom (𝑅↑_𝑟1))
122	119, 121	iunxsn 4918	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑥 ∈ {1}dom (𝑅↑_𝑟𝑥) = dom (𝑅↑_𝑟1)
123	118, 122	syl6eq 2849	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) = dom (𝑅↑_𝑟1))
124		relexp1g 14223	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑_𝑟1) = 𝑅)
125	124	ad2antll 725	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (𝑅↑_𝑟1) = 𝑅)
126	125	dmeqd 5667	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → dom (𝑅↑_𝑟1) = dom 𝑅)
127	123, 126	eqtrd 2833	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) = dom 𝑅)
128	117	iuneq1d 4857	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) = ∪ 𝑥 ∈ {1}ran (𝑅↑_𝑟𝑥))
129	120	rneqd 5697	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 1 → ran (𝑅↑_𝑟𝑥) = ran (𝑅↑_𝑟1))
130	119, 129	iunxsn 4918	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑥 ∈ {1}ran (𝑅↑_𝑟𝑥) = ran (𝑅↑_𝑟1)
131	128, 130	syl6eq 2849	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) = ran (𝑅↑_𝑟1))
132	125	rneqd 5697	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ran (𝑅↑_𝑟1) = ran 𝑅)
133	131, 132	eqtrd 2833	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) = ran 𝑅)
134	127, 133	uneq12d 4067	. . . . . . . . . . . . 13 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) = (dom 𝑅 ∪ ran 𝑅))
135	134	uneq2d 4066	. . . . . . . . . . . 12 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ((∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))) = ((∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ (dom 𝑅 ∪ ran 𝑅)))
136	88	ad2antll 725	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∀𝑥 ∈ ℕ₀ dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
137		olc 863	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑍 ⊆ ℕ₀ → ({0} ⊆ ℕ₀ ∨ 𝑍 ⊆ ℕ₀))
138	137	ad2antrl 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ({0} ⊆ ℕ₀ ∨ 𝑍 ⊆ ℕ₀))
139		inss 4141	. . . . . . . . . . . . . . . . . . . 20 ⊢ (({0} ⊆ ℕ₀ ∨ 𝑍 ⊆ ℕ₀) → ({0} ∩ 𝑍) ⊆ ℕ₀)
140	138, 139	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ({0} ∩ 𝑍) ⊆ ℕ₀)
141	140	sseld 3894	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (𝑥 ∈ ({0} ∩ 𝑍) → 𝑥 ∈ ℕ₀))
142	141	imim1d 82	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ((𝑥 ∈ ℕ₀ → dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)) → (𝑥 ∈ ({0} ∩ 𝑍) → dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))))
143	142	ralimdv2 3145	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (∀𝑥 ∈ ℕ₀ dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) → ∀𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
144	136, 143	mpd 15	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∀𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
145		iunss 4874	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
146	144, 145	sylibr 235	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
147	102	ad2antll 725	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∀𝑥 ∈ ℕ₀ ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
148	141	imim1d 82	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ((𝑥 ∈ ℕ₀ → ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)) → (𝑥 ∈ ({0} ∩ 𝑍) → ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))))
149	148	ralimdv2 3145	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (∀𝑥 ∈ ℕ₀ ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) → ∀𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
150	147, 149	mpd 15	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∀𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
151		iunss 4874	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
152	150, 151	sylibr 235	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
153	146, 152	unssd 4089	. . . . . . . . . . . . 13 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
154		ssequn1 4083	. . . . . . . . . . . . 13 ⊢ ((∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ((∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅))
155	153, 154	sylib 219	. . . . . . . . . . . 12 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ((∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅))
156	135, 155	eqtrd 2833	. . . . . . . . . . 11 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ((∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
157	156	ex 413	. . . . . . . . . 10 ⊢ (1 ∈ 𝑍 → ((𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉) → ((∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅)))
158	113, 157	jaoi 852	. . . . . . . . 9 ⊢ ((0 ∈ 𝑍 ∨ 1 ∈ 𝑍) → ((𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉) → ((∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅)))
159	57, 158	syl 17	. . . . . . . 8 ⊢ (({0, 1} ∩ 𝑍) ≠ ∅ → ((𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉) → ((∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅)))
160	159	3impib 1109	. . . . . . 7 ⊢ ((({0, 1} ∩ 𝑍) ≠ ∅ ∧ 𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉) → ((∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
161	160	3com13 1117	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀ ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ((∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
162	161	uneq1d 4065	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀ ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → (((∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))) ∪ (∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥))) = ((dom 𝑅 ∪ ran 𝑅) ∪ (∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥))))
163	88	adantr 481	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀) → ∀𝑥 ∈ ℕ₀ dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
164		ssel 3889	. . . . . . . . . . . . 13 ⊢ (𝑍 ⊆ ℕ₀ → (𝑥 ∈ 𝑍 → 𝑥 ∈ ℕ₀))
165	164	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀) → (𝑥 ∈ 𝑍 → 𝑥 ∈ ℕ₀))
166	165	imim1d 82	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀) → ((𝑥 ∈ ℕ₀ → dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)) → (𝑥 ∈ 𝑍 → dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))))
167	166	ralimdv2 3145	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀) → (∀𝑥 ∈ ℕ₀ dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) → ∀𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
168	163, 167	mpd 15	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀) → ∀𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
169		iunss 4874	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
170	168, 169	sylibr 235	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀) → ∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
171	102	adantr 481	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀) → ∀𝑥 ∈ ℕ₀ ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
172	165	imim1d 82	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀) → ((𝑥 ∈ ℕ₀ → ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)) → (𝑥 ∈ 𝑍 → ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))))
173	172	ralimdv2 3145	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀) → (∀𝑥 ∈ ℕ₀ ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) → ∀𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
174	171, 173	mpd 15	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀) → ∀𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
175		iunss 4874	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
176	174, 175	sylibr 235	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀) → ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
177	170, 176	unssd 4089	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀) → (∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
178	177	3adant3 1125	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀ ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → (∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
179		ssequn2 4086	. . . . . 6 ⊢ ((∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ((dom 𝑅 ∪ ran 𝑅) ∪ (∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
180	178, 179	sylib 219	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀ ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ((dom 𝑅 ∪ ran 𝑅) ∪ (∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
181	162, 180	eqtrd 2833	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀ ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → (((∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥)) ∪ (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))) ∪ (∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
182	35, 181	syl5eq 2845	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀ ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → (dom ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) ∪ ran ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥)) = (dom 𝑅 ∪ ran 𝑅))
183		nn0ex 11757	. . . . . . 7 ⊢ ℕ₀ ∈ V
184	183	ssex 5123	. . . . . 6 ⊢ (𝑍 ⊆ ℕ₀ → 𝑍 ∈ V)
185		incom 4105	. . . . . . . . . 10 ⊢ (𝑍 ∩ {0}) = ({0} ∩ 𝑍)
186		inex1g 5121	. . . . . . . . . 10 ⊢ (𝑍 ∈ V → (𝑍 ∩ {0}) ∈ V)
187	185, 186	syl5eqelr 2890	. . . . . . . . 9 ⊢ (𝑍 ∈ V → ({0} ∩ 𝑍) ∈ V)
188		incom 4105	. . . . . . . . . 10 ⊢ (𝑍 ∩ {1}) = ({1} ∩ 𝑍)
189		inex1g 5121	. . . . . . . . . 10 ⊢ (𝑍 ∈ V → (𝑍 ∩ {1}) ∈ V)
190	188, 189	syl5eqelr 2890	. . . . . . . . 9 ⊢ (𝑍 ∈ V → ({1} ∩ 𝑍) ∈ V)
191		unexg 7336	. . . . . . . . 9 ⊢ ((({0} ∩ 𝑍) ∈ V ∧ ({1} ∩ 𝑍) ∈ V) → (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∈ V)
192	187, 190, 191	syl2anc 584	. . . . . . . 8 ⊢ (𝑍 ∈ V → (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∈ V)
193		unexg 7336	. . . . . . . 8 ⊢ (((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∈ V ∧ 𝑍 ∈ V) → ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) ∈ V)
194	192, 193	mpancom 684	. . . . . . 7 ⊢ (𝑍 ∈ V → ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) ∈ V)
195		ovex 7055	. . . . . . . 8 ⊢ (𝑅↑_𝑟𝑥) ∈ V
196	195	rgenw 3119	. . . . . . 7 ⊢ ∀𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) ∈ V
197		iunexg 7527	. . . . . . 7 ⊢ ((((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) ∈ V ∧ ∀𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) ∈ V) → ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) ∈ V)
198	194, 196, 197	sylancl 586	. . . . . 6 ⊢ (𝑍 ∈ V → ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) ∈ V)
199	184, 198	syl 17	. . . . 5 ⊢ (𝑍 ⊆ ℕ₀ → ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) ∈ V)
200	199	3ad2ant2 1127	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀ ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) ∈ V)
201		simp1 1129	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀ ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → 𝑅 ∈ 𝑉)
202		relexp0eq 39552	. . . 4 ⊢ ((∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) ∈ V ∧ 𝑅 ∈ 𝑉) → ((dom ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) ∪ ran ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥)) = (dom 𝑅 ∪ ran 𝑅) ↔ (∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥)↑_𝑟0) = (𝑅↑_𝑟0)))
203	200, 201, 202	syl2anc 584	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀ ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ((dom ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) ∪ ran ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥)) = (dom 𝑅 ∪ ran 𝑅) ↔ (∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥)↑_𝑟0) = (𝑅↑_𝑟0)))
204	182, 203	mpbid 233	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀ ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → (∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥)↑_𝑟0) = (𝑅↑_𝑟0))
205	12, 204	syl5eq 2845	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀ ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → (∪ 𝑥 ∈ 𝑍 (𝑅↑_𝑟𝑥)↑_𝑟0) = (𝑅↑_𝑟0))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 842 ∧ w3a 1080 = wceq 1525 ∈ wcel 2083 ≠ wne 2986 ∀wral 3107 Vcvv 3440 ∪ cun 3863 ∩ cin 3864 ⊆ wss 3865 ∅c0 4217 {csn 4478 {cpr 4480 ∪ ciun 4831 I cid 5354 dom cdm 5450 ran crn 5451 ↾ cres 5452 (class class class)co 7023 0cc0 10390 1c1 10391 ℕ₀cn0 11751 ↑_𝑟crelexp 14217

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-n0 11752 df-z 11836 df-uz 12098 df-seq 13224 df-relexp 14218

This theorem is referenced by: corclrcl 39558 corcltrcl 39590

W3C validator