iunrelexp0 - Mathbox for Richard Penner

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Theorem iunrelexp0 44702

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 4586	. . . . . . 7 ⊢ {0, 1} = ({0} ∪ {1})
2	1	ineq1i 4113	. . . . . 6 ⊢ ({0, 1} ∩ 𝑍) = (({0} ∪ {1}) ∩ 𝑍)
3		indir 4231	. . . . . 6 ⊢ (({0} ∪ {1}) ∩ 𝑍) = (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))
4	2, 3	eqtr2i 2784	. . . . 5 ⊢ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) = ({0, 1} ∩ 𝑍)
5	4	uneq1i 4163	. . . 4 ⊢ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) = (({0, 1} ∩ 𝑍) ∪ 𝑍)
6		inss2 4134	. . . . 5 ⊢ ({0, 1} ∩ 𝑍) ⊆ 𝑍
7		ssequn1 4184	. . . . 5 ⊢ (({0, 1} ∩ 𝑍) ⊆ 𝑍 ↔ (({0, 1} ∩ 𝑍) ∪ 𝑍) = 𝑍)
8	6, 7	mpbi 233	. . . 4 ⊢ (({0, 1} ∩ 𝑍) ∪ 𝑍) = 𝑍
9	5, 8	eqtr2i 2784	. . 3 ⊢ 𝑍 = ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)
10		iuneq1 4967	. . . 4 ⊢ (𝑍 = ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) → ∪ 𝑥 ∈ 𝑍 (𝑅↑_𝑟𝑥) = ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥))
11	10	oveq1d 7435	. . 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 5894	. . . . . . 7 ⊢ dom ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) = ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)dom (𝑅↑_𝑟𝑥)
14		iunxun 5053	. . . . . . 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 5053	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅↑_𝑟𝑥) = (∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥))
16	15	equncomi 4159	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅↑_𝑟𝑥) = (∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥))
17	16	uneq1i 4163	. . . . . . . 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 4159	. . . . . . 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 2787	. . . . . 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 6138	. . . . . . 7 ⊢ ran ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) = ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)ran (𝑅↑_𝑟𝑥)
21		iunxun 5053	. . . . . . 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 5053	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))ran (𝑅↑_𝑟𝑥) = (∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥))
23	22	uneq1i 4163	. . . . . . 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 2787	. . . . . 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 4165	. . . . 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 4157	. . . . . . 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 4163	. . . . . 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 4173	. . . . . 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 2783	. . . . 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 4157	. . . . . . . 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 4163	. . . . . . 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 4173	. . . . . . 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 2783	. . . . . 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 4163	. . . . 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 2787	. . . 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 2956	. . . . . . . . . 10 ⊢ (({0, 1} ∩ 𝑍) ≠ ∅ ↔ ¬ ({0, 1} ∩ 𝑍) = ∅)
37		incom 4106	. . . . . . . . . . . . . . 15 ⊢ ({0, 1} ∩ 𝑍) = (𝑍 ∩ {0, 1})
38	1	ineq2i 4114	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∩ {0, 1}) = (𝑍 ∩ ({0} ∪ {1}))
39		indi 4229	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∩ ({0} ∪ {1})) = ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1}))
40	37, 38, 39	3eqtri 2787	. . . . . . . . . . . . . 14 ⊢ ({0, 1} ∩ 𝑍) = ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1}))
41	40	eqeq1i 2765	. . . . . . . . . . . . 13 ⊢ (({0, 1} ∩ 𝑍) = ∅ ↔ ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1})) = ∅)
42		un00 4356	. . . . . . . . . . . . 13 ⊢ (((𝑍 ∩ {0}) = ∅ ∧ (𝑍 ∩ {1}) = ∅) ↔ ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1})) = ∅)
43		anor 998	. . . . . . . . . . . . 13 ⊢ (((𝑍 ∩ {0}) = ∅ ∧ (𝑍 ∩ {1}) = ∅) ↔ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
44	41, 42, 43	3bitr2i 302	. . . . . . . . . . . 12 ⊢ (({0, 1} ∩ 𝑍) = ∅ ↔ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
45	44	notbii 323	. . . . . . . . . . 11 ⊢ (¬ ({0, 1} ∩ 𝑍) = ∅ ↔ ¬ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
46		notnotb 318	. . . . . . . . . . 11 ⊢ ((¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅) ↔ ¬ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
47		disjsn 4671	. . . . . . . . . . . . . 14 ⊢ ((𝑍 ∩ {0}) = ∅ ↔ ¬ 0 ∈ 𝑍)
48	47	notbii 323	. . . . . . . . . . . . 13 ⊢ (¬ (𝑍 ∩ {0}) = ∅ ↔ ¬ ¬ 0 ∈ 𝑍)
49		notnotb 318	. . . . . . . . . . . . 13 ⊢ (0 ∈ 𝑍 ↔ ¬ ¬ 0 ∈ 𝑍)
50	48, 49	bitr4i 281	. . . . . . . . . . . 12 ⊢ (¬ (𝑍 ∩ {0}) = ∅ ↔ 0 ∈ 𝑍)
51		disjsn 4671	. . . . . . . . . . . . . 14 ⊢ ((𝑍 ∩ {1}) = ∅ ↔ ¬ 1 ∈ 𝑍)
52	51	notbii 323	. . . . . . . . . . . . 13 ⊢ (¬ (𝑍 ∩ {1}) = ∅ ↔ ¬ ¬ 1 ∈ 𝑍)
53		notnotb 318	. . . . . . . . . . . . 13 ⊢ (1 ∈ 𝑍 ↔ ¬ ¬ 1 ∈ 𝑍)
54	52, 53	bitr4i 281	. . . . . . . . . . . 12 ⊢ (¬ (𝑍 ∩ {1}) = ∅ ↔ 1 ∈ 𝑍)
55	50, 54	orbi12i 928	. . . . . . . . . . 11 ⊢ ((¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅) ↔ (0 ∈ 𝑍 ∨ 1 ∈ 𝑍))
56	45, 46, 55	3bitr2i 302	. . . . . . . . . 10 ⊢ (¬ ({0, 1} ∩ 𝑍) = ∅ ↔ (0 ∈ 𝑍 ∨ 1 ∈ 𝑍))
57	36, 56	sylbb 222	. . . . . . . . 9 ⊢ (({0, 1} ∩ 𝑍) ≠ ∅ → (0 ∈ 𝑍 ∨ 1 ∈ 𝑍))
58		simpl 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → 0 ∈ 𝑍)
59	58	snssd 4746	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → {0} ⊆ 𝑍)
60		dfss2 4094	. . . . . . . . . . . . . . . . . . 19 ⊢ ({0} ⊆ 𝑍 ↔ ({0} ∩ 𝑍) = {0})
61	59, 60	sylib 221	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ({0} ∩ 𝑍) = {0})
62	61	iuneq1d 4978	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) = ∪ 𝑥 ∈ {0}dom (𝑅↑_𝑟𝑥))
63		c0ex 11300	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
64		oveq2 7428	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 0 → (𝑅↑_𝑟𝑥) = (𝑅↑_𝑟0))
65	64	dmeqd 5886	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 0 → dom (𝑅↑_𝑟𝑥) = dom (𝑅↑_𝑟0))
66	63, 65	iunxsn 5050	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑥 ∈ {0}dom (𝑅↑_𝑟𝑥) = dom (𝑅↑_𝑟0)
67	62, 66	eqtrdi 2811	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) = dom (𝑅↑_𝑟0))
68		relexp0g 15175	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑_𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
69	68	ad2antll 742	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (𝑅↑_𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
70	69	dmeqd 5886	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → dom (𝑅↑_𝑟0) = dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
71		dmresi 6043	. . . . . . . . . . . . . . . . 17 ⊢ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
72	70, 71	eqtrdi 2811	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → dom (𝑅↑_𝑟0) = (dom 𝑅 ∪ ran 𝑅))
73	67, 72	eqtrd 2795	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) = (dom 𝑅 ∪ ran 𝑅))
74	61	iuneq1d 4978	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) = ∪ 𝑥 ∈ {0}ran (𝑅↑_𝑟𝑥))
75	64	rneqd 5919	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 0 → ran (𝑅↑_𝑟𝑥) = ran (𝑅↑_𝑟0))
76	63, 75	iunxsn 5050	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑥 ∈ {0}ran (𝑅↑_𝑟𝑥) = ran (𝑅↑_𝑟0)
77	74, 76	eqtrdi 2811	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) = ran (𝑅↑_𝑟0))
78	69	rneqd 5919	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ran (𝑅↑_𝑟0) = ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
79		rnresi 6072	. . . . . . . . . . . . . . . . 17 ⊢ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
80	78, 79	eqtrdi 2811	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ran (𝑅↑_𝑟0) = (dom 𝑅 ∪ ran 𝑅))
81	77, 80	eqtrd 2795	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) = (dom 𝑅 ∪ ran 𝑅))
82	73, 81	uneq12d 4168	. . . . . . . . . . . . . 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 4156	. . . . . . . . . . . . . 14 ⊢ ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
84	82, 83	eqtrdi 2811	. . . . . . . . . . . . 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 4166	. . . . . . . . . . . 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 15195	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
87	86	expcom 419	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ ℕ₀ → dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
88	87	ralrimiv 3153	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ 𝑉 → ∀𝑥 ∈ ℕ₀ dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
89	88	ad2antll 742	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∀𝑥 ∈ ℕ₀ dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
90		olc 882	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑍 ⊆ ℕ₀ → ({1} ⊆ ℕ₀ ∨ 𝑍 ⊆ ℕ₀))
91	90	ad2antrl 741	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ({1} ⊆ ℕ₀ ∨ 𝑍 ⊆ ℕ₀))
92		inss 4145	. . . . . . . . . . . . . . . . . . . 20 ⊢ (({1} ⊆ ℕ₀ ∨ 𝑍 ⊆ ℕ₀) → ({1} ∩ 𝑍) ⊆ ℕ₀)
93	91, 92	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ({1} ∩ 𝑍) ⊆ ℕ₀)
94	93	sseld 3907	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (𝑥 ∈ ({1} ∩ 𝑍) → 𝑥 ∈ ℕ₀))
95	94	imim1d 83	. . . . . . . . . . . . . . . . 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 3171	. . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∀𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
98		iunss 5002	. . . . . . . . . . . . . . 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 237	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
100		relexprng 15199	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
101	100	expcom 419	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ ℕ₀ → ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
102	101	ralrimiv 3153	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ 𝑉 → ∀𝑥 ∈ ℕ₀ ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
103	102	ad2antll 742	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∀𝑥 ∈ ℕ₀ ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
104	94	imim1d 83	. . . . . . . . . . . . . . . . 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 3171	. . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∀𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
107		iunss 5002	. . . . . . . . . . . . . . 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 237	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
109	99, 108	unssd 4190	. . . . . . . . . . . . 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 4187	. . . . . . . . . . . . 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 221	. . . . . . . . . . . 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 2795	. . . . . . . . . . 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 418	. . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → 1 ∈ 𝑍)
115	114	snssd 4746	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → {1} ⊆ 𝑍)
116		dfss2 4094	. . . . . . . . . . . . . . . . . 18 ⊢ ({1} ⊆ 𝑍 ↔ ({1} ∩ 𝑍) = {1})
117	115, 116	sylib 221	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ({1} ∩ 𝑍) = {1})
118	117	iuneq1d 4978	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) = ∪ 𝑥 ∈ {1}dom (𝑅↑_𝑟𝑥))
119		1ex 11303	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ V
120		oveq2 7428	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 1 → (𝑅↑_𝑟𝑥) = (𝑅↑_𝑟1))
121	120	dmeqd 5886	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 1 → dom (𝑅↑_𝑟𝑥) = dom (𝑅↑_𝑟1))
122	119, 121	iunxsn 5050	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑥 ∈ {1}dom (𝑅↑_𝑟𝑥) = dom (𝑅↑_𝑟1)
123	118, 122	eqtrdi 2811	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) = dom (𝑅↑_𝑟1))
124		relexp1g 15179	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑_𝑟1) = 𝑅)
125	124	ad2antll 742	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (𝑅↑_𝑟1) = 𝑅)
126	125	dmeqd 5886	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → dom (𝑅↑_𝑟1) = dom 𝑅)
127	123, 126	eqtrd 2795	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) = dom 𝑅)
128	117	iuneq1d 4978	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) = ∪ 𝑥 ∈ {1}ran (𝑅↑_𝑟𝑥))
129	120	rneqd 5919	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 1 → ran (𝑅↑_𝑟𝑥) = ran (𝑅↑_𝑟1))
130	119, 129	iunxsn 5050	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑥 ∈ {1}ran (𝑅↑_𝑟𝑥) = ran (𝑅↑_𝑟1)
131	128, 130	eqtrdi 2811	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) = ran (𝑅↑_𝑟1))
132	125	rneqd 5919	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ran (𝑅↑_𝑟1) = ran 𝑅)
133	131, 132	eqtrd 2795	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) = ran 𝑅)
134	127, 133	uneq12d 4168	. . . . . . . . . . . . 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 4167	. . . . . . . . . . . 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 742	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∀𝑥 ∈ ℕ₀ dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
137		olc 882	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑍 ⊆ ℕ₀ → ({0} ⊆ ℕ₀ ∨ 𝑍 ⊆ ℕ₀))
138	137	ad2antrl 741	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ({0} ⊆ ℕ₀ ∨ 𝑍 ⊆ ℕ₀))
139		inss 4145	. . . . . . . . . . . . . . . . . . . 20 ⊢ (({0} ⊆ ℕ₀ ∨ 𝑍 ⊆ ℕ₀) → ({0} ∩ 𝑍) ⊆ ℕ₀)
140	138, 139	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ({0} ∩ 𝑍) ⊆ ℕ₀)
141	140	sseld 3907	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (𝑥 ∈ ({0} ∩ 𝑍) → 𝑥 ∈ ℕ₀))
142	141	imim1d 83	. . . . . . . . . . . . . . . . 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 3171	. . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∀𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
145		iunss 5002	. . . . . . . . . . . . . . 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 237	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
147	102	ad2antll 742	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∀𝑥 ∈ ℕ₀ ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
148	141	imim1d 83	. . . . . . . . . . . . . . . . 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 3171	. . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∀𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
151		iunss 5002	. . . . . . . . . . . . . . 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 237	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
153	146, 152	unssd 4190	. . . . . . . . . . . . 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 4184	. . . . . . . . . . . . 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 221	. . . . . . . . . . . 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 2795	. . . . . . . . . . 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 418	. . . . . . . . . 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 871	. . . . . . . . 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 18	. . . . . . . 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 1134	. . . . . . 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 1142	. . . . . 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 4166	. . . . 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 486	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀) → ∀𝑥 ∈ ℕ₀ dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
164		ssel 3902	. . . . . . . . . . . . 13 ⊢ (𝑍 ⊆ ℕ₀ → (𝑥 ∈ 𝑍 → 𝑥 ∈ ℕ₀))
165	164	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀) → (𝑥 ∈ 𝑍 → 𝑥 ∈ ℕ₀))
166	165	imim1d 83	. . . . . . . . . . 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 3171	. . . . . . . . . 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 16	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀) → ∀𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
169		iunss 5002	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
170	168, 169	sylibr 237	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀) → ∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
171	102	adantr 486	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀) → ∀𝑥 ∈ ℕ₀ ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
172	165	imim1d 83	. . . . . . . . . . 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 3171	. . . . . . . . . 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 16	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀) → ∀𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
175		iunss 5002	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
176	174, 175	sylibr 237	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀) → ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
177	170, 176	unssd 4190	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀) → (∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
178	177	3adant3 1150	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀ ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → (∪ 𝑥 ∈ 𝑍 dom (𝑅↑_𝑟𝑥) ∪ ∪ 𝑥 ∈ 𝑍 ran (𝑅↑_𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
179		ssequn2 4187	. . . . . 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 221	. . . . 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 2795	. . . 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	eqtrid 2807	. . 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 12612	. . . . . . 7 ⊢ ℕ₀ ∈ V
184	183	ssex 5281	. . . . . 6 ⊢ (𝑍 ⊆ ℕ₀ → 𝑍 ∈ V)
185		inex2g 5279	. . . . . . . . 9 ⊢ (𝑍 ∈ V → ({0} ∩ 𝑍) ∈ V)
186		inex2g 5279	. . . . . . . . 9 ⊢ (𝑍 ∈ V → ({1} ∩ 𝑍) ∈ V)
187	185, 186	unexd 7768	. . . . . . . 8 ⊢ (𝑍 ∈ V → (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∈ V)
188		unexg 7760	. . . . . . . 8 ⊢ (((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∈ V ∧ 𝑍 ∈ V) → ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) ∈ V)
189	187, 188	mpancom 701	. . . . . . 7 ⊢ (𝑍 ∈ V → ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) ∈ V)
190		ovex 7453	. . . . . . . 8 ⊢ (𝑅↑_𝑟𝑥) ∈ V
191	190	rgenw 3080	. . . . . . 7 ⊢ ∀𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) ∈ V
192		iunexg 7975	. . . . . . 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)
193	189, 191, 192	sylancl 598	. . . . . 6 ⊢ (𝑍 ∈ V → ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) ∈ V)
194	184, 193	syl 18	. . . . 5 ⊢ (𝑍 ⊆ ℕ₀ → ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) ∈ V)
195	194	3ad2ant2 1152	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀ ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥) ∈ V)
196		simp1 1154	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀ ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → 𝑅 ∈ 𝑉)
197		relexp0eq 44701	. . . 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)))
198	195, 196, 197	syl2anc 596	. . 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)))
199	182, 198	mpbid 235	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀ ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → (∪ 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅↑_𝑟𝑥)↑_𝑟0) = (𝑅↑_𝑟0))
200	12, 199	eqtrid 2807	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ₀ ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → (∪ 𝑥 ∈ 𝑍 (𝑅↑_𝑟𝑥)↑_𝑟0) = (𝑅↑_𝑟0))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 Vcvv 3450 ⊆ wss 3895 ∩ cin 4089 ∪ cun 4149 ∅c0 4278 {csn 4583 {cpr 4585 ∪ ciun 4950 I cid 5544 dom cdm 5650 ran crn 5651 ↾ cres 5652 (class class class)co 7420 0cc0 11200 1c1 11201 ℕ₀cn0 12606 ↑_𝑟crelexp 15172

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-er 8717 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-n0 12607 df-z 12694 df-uz 12966 df-seq 14145 df-relexp 15173

This theorem is used by: corclrcl 44707 corcltrcl 44739

W3C validator