iunrelexp0 - Mathbox for Richard Penner

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Theorem iunrelexp0 37994

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 4180	. . . . . . 7 ⊢ {0, 1} = ({0} ∪ {1})
2	1	ineq1i 3810	. . . . . 6 ⊢ ({0, 1} ∩ 𝑍) = (({0} ∪ {1}) ∩ 𝑍)
3		indir 3875	. . . . . 6 ⊢ (({0} ∪ {1}) ∩ 𝑍) = (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))
4	2, 3	eqtr2i 2645	. . . . 5 ⊢ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) = ({0, 1} ∩ 𝑍)
5	4	uneq1i 3763	. . . 4 ⊢ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) = (({0, 1} ∩ 𝑍) ∪ 𝑍)
6		inss2 3834	. . . . 5 ⊢ ({0, 1} ∩ 𝑍) ⊆ 𝑍
7		ssequn1 3783	. . . . 5 ⊢ (({0, 1} ∩ 𝑍) ⊆ 𝑍 ↔ (({0, 1} ∩ 𝑍) ∪ 𝑍) = 𝑍)
8	6, 7	mpbi 220	. . . 4 ⊢ (({0, 1} ∩ 𝑍) ∪ 𝑍) = 𝑍
9	5, 8	eqtr2i 2645	. . 3 ⊢ 𝑍 = ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)
10		iuneq1 4534	. . . 4 ⊢ (𝑍 = ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) → ∪ 𝑥 ∈ 𝑍 (𝑅↑_𝑟𝑥) = ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥))
11	10	oveq1d 6665	. . 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 5333	. . . . . . 7 ⊢ dom ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) = ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)dom (𝑅↑_𝑟𝑥)
14		iunxun 4605	. . . . . . 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 4605	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅↑_𝑟𝑥) = (∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥))
16	15	equncomi 3759	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅↑_𝑟𝑥) = (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥))
17	16	uneq1i 3763	. . . . . . . 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 3759	. . . . . . 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 2648	. . . . . 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 5543	. . . . . . 7 ⊢ ran ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) = ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)ran (𝑅↑_𝑟𝑥)
21		iunxun 4605	. . . . . . 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 4605	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))ran (𝑅↑_𝑟𝑥) = (∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))
23	22	uneq1i 3763	. . . . . . 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 2648	. . . . . 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 3765	. . . . 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 3757	. . . . . . 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 3763	. . . . . 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 3773	. . . . . 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 2644	. . . . 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 3757	. . . . . . . 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 3763	. . . . . . 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 3773	. . . . . . 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 2644	. . . . . 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 3763	. . . . 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 2648	. . . 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 2795	. . . . . . . . . 10 ⊢ (({0, 1} ∩ 𝑍) ≠ ∅ ↔ ¬ ({0, 1} ∩ 𝑍) = ∅)
37		incom 3805	. . . . . . . . . . . . . . 15 ⊢ ({0, 1} ∩ 𝑍) = (𝑍 ∩ {0, 1})
38	1	ineq2i 3811	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∩ {0, 1}) = (𝑍 ∩ ({0} ∪ {1}))
39		indi 3873	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∩ ({0} ∪ {1})) = ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1}))
40	37, 38, 39	3eqtri 2648	. . . . . . . . . . . . . 14 ⊢ ({0, 1} ∩ 𝑍) = ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1}))
41	40	eqeq1i 2627	. . . . . . . . . . . . 13 ⊢ (({0, 1} ∩ 𝑍) = ∅ ↔ ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1})) = ∅)
42		un00 4011	. . . . . . . . . . . . 13 ⊢ (((𝑍 ∩ {0}) = ∅ ∧ (𝑍 ∩ {1}) = ∅) ↔ ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1})) = ∅)
43		anor 510	. . . . . . . . . . . . 13 ⊢ (((𝑍 ∩ {0}) = ∅ ∧ (𝑍 ∩ {1}) = ∅) ↔ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
44	41, 42, 43	3bitr2i 288	. . . . . . . . . . . 12 ⊢ (({0, 1} ∩ 𝑍) = ∅ ↔ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
45	44	notbii 310	. . . . . . . . . . 11 ⊢ (¬ ({0, 1} ∩ 𝑍) = ∅ ↔ ¬ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
46		notnotb 304	. . . . . . . . . . 11 ⊢ ((¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅) ↔ ¬ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
47		disjsn 4246	. . . . . . . . . . . . . 14 ⊢ ((𝑍 ∩ {0}) = ∅ ↔ ¬ 0 ∈ 𝑍)
48	47	notbii 310	. . . . . . . . . . . . 13 ⊢ (¬ (𝑍 ∩ {0}) = ∅ ↔ ¬ ¬ 0 ∈ 𝑍)
49		notnotb 304	. . . . . . . . . . . . 13 ⊢ (0 ∈ 𝑍 ↔ ¬ ¬ 0 ∈ 𝑍)
50	48, 49	bitr4i 267	. . . . . . . . . . . 12 ⊢ (¬ (𝑍 ∩ {0}) = ∅ ↔ 0 ∈ 𝑍)
51		disjsn 4246	. . . . . . . . . . . . . 14 ⊢ ((𝑍 ∩ {1}) = ∅ ↔ ¬ 1 ∈ 𝑍)
52	51	notbii 310	. . . . . . . . . . . . 13 ⊢ (¬ (𝑍 ∩ {1}) = ∅ ↔ ¬ ¬ 1 ∈ 𝑍)
53		notnotb 304	. . . . . . . . . . . . 13 ⊢ (1 ∈ 𝑍 ↔ ¬ ¬ 1 ∈ 𝑍)
54	52, 53	bitr4i 267	. . . . . . . . . . . 12 ⊢ (¬ (𝑍 ∩ {1}) = ∅ ↔ 1 ∈ 𝑍)
55	50, 54	orbi12i 543	. . . . . . . . . . 11 ⊢ ((¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅) ↔ (0 ∈ 𝑍 ∨ 1 ∈ 𝑍))
56	45, 46, 55	3bitr2i 288	. . . . . . . . . 10 ⊢ (¬ ({0, 1} ∩ 𝑍) = ∅ ↔ (0 ∈ 𝑍 ∨ 1 ∈ 𝑍))
57	36, 56	sylbb 209	. . . . . . . . 9 ⊢ (({0, 1} ∩ 𝑍) ≠ ∅ → (0 ∈ 𝑍 ∨ 1 ∈ 𝑍))
58		simpl 473	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → 0 ∈ 𝑍)
59	58	snssd 4340	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → {0} ⊆ 𝑍)
60		df-ss 3588	. . . . . . . . . . . . . . . . . . 19 ⊢ ({0} ⊆ 𝑍 ↔ ({0} ∩ 𝑍) = {0})
61	59, 60	sylib 208	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ({0} ∩ 𝑍) = {0})
62	61	iuneq1d 4545	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) = ∪ 𝑥 ∈ {0}dom (𝑅↑_𝑟𝑥))
63		c0ex 10034	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
64		oveq2 6658	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 0 → (𝑅↑_𝑟𝑥) = (𝑅↑_𝑟0))
65	64	dmeqd 5326	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 0 → dom (𝑅↑_𝑟𝑥) = dom (𝑅↑_𝑟0))
66	63, 65	iunxsn 4603	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑥 ∈ {0}dom (𝑅↑_𝑟𝑥) = dom (𝑅↑_𝑟0)
67	62, 66	syl6eq 2672	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) = dom (𝑅↑_𝑟0))
68		relexp0g 13762	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑_𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
69	68	ad2antll 765	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (𝑅↑_𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
70	69	dmeqd 5326	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → dom (𝑅↑_𝑟0) = dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
71		dmresi 5457	. . . . . . . . . . . . . . . . 17 ⊢ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
72	70, 71	syl6eq 2672	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → dom (𝑅↑_𝑟0) = (dom 𝑅 ∪ ran 𝑅))
73	67, 72	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) = (dom 𝑅 ∪ ran 𝑅))
74	61	iuneq1d 4545	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) = ∪ 𝑥 ∈ {0}ran (𝑅↑_𝑟𝑥))
75	64	rneqd 5353	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 0 → ran (𝑅↑_𝑟𝑥) = ran (𝑅↑_𝑟0))
76	63, 75	iunxsn 4603	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑥 ∈ {0}ran (𝑅↑_𝑟𝑥) = ran (𝑅↑_𝑟0)
77	74, 76	syl6eq 2672	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) = ran (𝑅↑_𝑟0))
78	69	rneqd 5353	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ran (𝑅↑_𝑟0) = ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
79		rnresi 5479	. . . . . . . . . . . . . . . . 17 ⊢ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
80	78, 79	syl6eq 2672	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ran (𝑅↑_𝑟0) = (dom 𝑅 ∪ ran 𝑅))
81	77, 80	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) = (dom 𝑅 ∪ ran 𝑅))
82	73, 81	uneq12d 3768	. . . . . . . . . . . . . 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 3756	. . . . . . . . . . . . . 14 ⊢ ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
84	82, 83	syl6eq 2672	. . . . . . . . . . . . 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 3766	. . . . . . . . . . . 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 13782	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
87	86	expcom 451	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ ℕ₀ → dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
88	87	ralrimiv 2965	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ 𝑉 → ∀𝑥 ∈ ℕ₀ dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
89	88	ad2antll 765	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∀𝑥 ∈ ℕ₀ dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
90		olc 399	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑍 ⊆ ℕ₀ → ({1} ⊆ ℕ₀ ∨ 𝑍 ⊆ ℕ₀))
91	90	ad2antrl 764	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ({1} ⊆ ℕ₀ ∨ 𝑍 ⊆ ℕ₀))
92		inss 3842	. . . . . . . . . . . . . . . . . . . 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 3602	. . . . . . . . . . . . . . . . . 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 2961	. . . . . . . . . . . . . . . 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 4561	. . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
100		relexprng 13786	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
101	100	expcom 451	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ ℕ₀ → ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
102	101	ralrimiv 2965	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ 𝑉 → ∀𝑥 ∈ ℕ₀ ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
103	102	ad2antll 765	. . . . . . . . . . . . . . . 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 2961	. . . . . . . . . . . . . . . 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 4561	. . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
109	99, 108	unssd 3789	. . . . . . . . . . . . 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 3786	. . . . . . . . . . . . 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 208	. . . . . . . . . . . 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 2656	. . . . . . . . . . 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 450	. . . . . . . . . 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 473	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → 1 ∈ 𝑍)
115	114	snssd 4340	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → {1} ⊆ 𝑍)
116		df-ss 3588	. . . . . . . . . . . . . . . . . 18 ⊢ ({1} ⊆ 𝑍 ↔ ({1} ∩ 𝑍) = {1})
117	115, 116	sylib 208	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ({1} ∩ 𝑍) = {1})
118	117	iuneq1d 4545	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) = ∪ 𝑥 ∈ {1}dom (𝑅↑_𝑟𝑥))
119		1ex 10035	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ V
120		oveq2 6658	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 1 → (𝑅↑_𝑟𝑥) = (𝑅↑_𝑟1))
121	120	dmeqd 5326	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 1 → dom (𝑅↑_𝑟𝑥) = dom (𝑅↑_𝑟1))
122	119, 121	iunxsn 4603	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑥 ∈ {1}dom (𝑅↑_𝑟𝑥) = dom (𝑅↑_𝑟1)
123	118, 122	syl6eq 2672	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) = dom (𝑅↑_𝑟1))
124		relexp1g 13766	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑_𝑟1) = 𝑅)
125	124	ad2antll 765	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (𝑅↑_𝑟1) = 𝑅)
126	125	dmeqd 5326	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → dom (𝑅↑_𝑟1) = dom 𝑅)
127	123, 126	eqtrd 2656	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) = dom 𝑅)
128	117	iuneq1d 4545	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) = ∪ 𝑥 ∈ {1}ran (𝑅↑_𝑟𝑥))
129	120	rneqd 5353	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 1 → ran (𝑅↑_𝑟𝑥) = ran (𝑅↑_𝑟1))
130	119, 129	iunxsn 4603	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑥 ∈ {1}ran (𝑅↑_𝑟𝑥) = ran (𝑅↑_𝑟1)
131	128, 130	syl6eq 2672	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) = ran (𝑅↑_𝑟1))
132	125	rneqd 5353	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ran (𝑅↑_𝑟1) = ran 𝑅)
133	131, 132	eqtrd 2656	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) = ran 𝑅)
134	127, 133	uneq12d 3768	. . . . . . . . . . . . 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 3767	. . . . . . . . . . . 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 765	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∀𝑥 ∈ ℕ₀ dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
137		olc 399	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑍 ⊆ ℕ₀ → ({0} ⊆ ℕ₀ ∨ 𝑍 ⊆ ℕ₀))
138	137	ad2antrl 764	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ({0} ⊆ ℕ₀ ∨ 𝑍 ⊆ ℕ₀))
139		inss 3842	. . . . . . . . . . . . . . . . . . . 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 3602	. . . . . . . . . . . . . . . . . 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 2961	. . . . . . . . . . . . . . . 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 4561	. . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
147	102	ad2antll 765	. . . . . . . . . . . . . . . 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 2961	. . . . . . . . . . . . . . . 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 4561	. . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
153	146, 152	unssd 3789	. . . . . . . . . . . . 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 3783	. . . . . . . . . . . . 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 208	. . . . . . . . . . . 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 2656	. . . . . . . . . . 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 450	. . . . . . . . . 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 394	. . . . . . . . 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 1262	. . . . . . 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 1270	. . . . . 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 3766	. . . . 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 3597	. . . . . . . . . . . . 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 2961	. . . . . . . . . 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 4561	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
170	168, 169	sylibr 224	. . . . . . . 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 2961	. . . . . . . . . 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 4561	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
176	174, 175	sylibr 224	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀) → ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
177	170, 176	unssd 3789	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀) → (∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
178	177	3adant3 1081	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀ ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → (∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
179		ssequn2 3786	. . . . . 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 208	. . . . 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 2656	. . . 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 2668	. . 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 11298	. . . . . . 7 ⊢ ℕ₀ ∈ V
184	183	ssex 4802	. . . . . 6 ⊢ (𝑍 ⊆ ℕ₀ → 𝑍 ∈ V)
185		incom 3805	. . . . . . . . . 10 ⊢ (𝑍 ∩ {0}) = ({0} ∩ 𝑍)
186		inex1g 4801	. . . . . . . . . 10 ⊢ (𝑍 ∈ V → (𝑍 ∩ {0}) ∈ V)
187	185, 186	syl5eqelr 2706	. . . . . . . . 9 ⊢ (𝑍 ∈ V → ({0} ∩ 𝑍) ∈ V)
188		incom 3805	. . . . . . . . . 10 ⊢ (𝑍 ∩ {1}) = ({1} ∩ 𝑍)
189		inex1g 4801	. . . . . . . . . 10 ⊢ (𝑍 ∈ V → (𝑍 ∩ {1}) ∈ V)
190	188, 189	syl5eqelr 2706	. . . . . . . . 9 ⊢ (𝑍 ∈ V → ({1} ∩ 𝑍) ∈ V)
191		unexg 6959	. . . . . . . . 9 ⊢ ((({0} ∩ 𝑍) ∈ V ∧ ({1} ∩ 𝑍) ∈ V) → (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∈ V)
192	187, 190, 191	syl2anc 693	. . . . . . . 8 ⊢ (𝑍 ∈ V → (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∈ V)
193		unexg 6959	. . . . . . . 8 ⊢ (((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∈ V ∧ 𝑍 ∈ V) → ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) ∈ V)
194	192, 193	mpancom 703	. . . . . . 7 ⊢ (𝑍 ∈ V → ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) ∈ V)
195		ovex 6678	. . . . . . . 8 ⊢ (𝑅↑_𝑟𝑥) ∈ V
196	195	rgenw 2924	. . . . . . 7 ⊢ ∀𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) ∈ V
197		iunexg 7143	. . . . . . 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 694	. . . . . 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 1083	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀ ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) ∈ V)
201		simp1 1061	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀ ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → 𝑅 ∈ 𝑉)
202		relexp0eq 37993	. . . 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 693	. . 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 222	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀ ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → (∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥)↑_𝑟0) = (𝑅↑_𝑟0))
205	12, 204	syl5eq 2668	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 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 Vcvv 3200 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 {csn 4177 {cpr 4179 ∪ ciun 4520 I cid 5023 dom cdm 5114 ran crn 5115 ↾ cres 5116 (class class class)co 6650 0cc0 9936 1c1 9937 ℕ₀cn0 11292 ↑_𝑟crelexp 13760

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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-seq 12802 df-relexp 13761

This theorem is referenced by: corclrcl 37999 corcltrcl 38031

W3C validator