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Theorem iunrelexp0 42048
Description: Simplification of zeroth power of indexed union of powers of relations. (Contributed by RP, 19-Jun-2020.)
Assertion
Ref Expression
iunrelexp0 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ( 𝑥𝑍 (𝑅𝑟𝑥)↑𝑟0) = (𝑅𝑟0))
Distinct variable groups:   𝑥,𝑅   𝑥,𝑉   𝑥,𝑍

Proof of Theorem iunrelexp0
StepHypRef Expression
1 df-pr 4594 . . . . . . 7 {0, 1} = ({0} ∪ {1})
21ineq1i 4173 . . . . . 6 ({0, 1} ∩ 𝑍) = (({0} ∪ {1}) ∩ 𝑍)
3 indir 4240 . . . . . 6 (({0} ∪ {1}) ∩ 𝑍) = (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))
42, 3eqtr2i 2766 . . . . 5 (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) = ({0, 1} ∩ 𝑍)
54uneq1i 4124 . . . 4 ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) = (({0, 1} ∩ 𝑍) ∪ 𝑍)
6 inss2 4194 . . . . 5 ({0, 1} ∩ 𝑍) ⊆ 𝑍
7 ssequn1 4145 . . . . 5 (({0, 1} ∩ 𝑍) ⊆ 𝑍 ↔ (({0, 1} ∩ 𝑍) ∪ 𝑍) = 𝑍)
86, 7mpbi 229 . . . 4 (({0, 1} ∩ 𝑍) ∪ 𝑍) = 𝑍
95, 8eqtr2i 2766 . . 3 𝑍 = ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)
10 iuneq1 4975 . . . 4 (𝑍 = ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) → 𝑥𝑍 (𝑅𝑟𝑥) = 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥))
1110oveq1d 7377 . . 3 (𝑍 = ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) → ( 𝑥𝑍 (𝑅𝑟𝑥)↑𝑟0) = ( 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥)↑𝑟0))
129, 11ax-mp 5 . 2 ( 𝑥𝑍 (𝑅𝑟𝑥)↑𝑟0) = ( 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥)↑𝑟0)
13 dmiun 5874 . . . . . . 7 dom 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) = 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)dom (𝑅𝑟𝑥)
14 iunxun 5059 . . . . . . 7 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)dom (𝑅𝑟𝑥) = ( 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 dom (𝑅𝑟𝑥))
15 iunxun 5059 . . . . . . . . . 10 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅𝑟𝑥) = ( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥))
1615equncomi 4120 . . . . . . . . 9 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅𝑟𝑥) = ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥))
1716uneq1i 4124 . . . . . . . 8 ( 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 dom (𝑅𝑟𝑥)) = (( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)) ∪ 𝑥𝑍 dom (𝑅𝑟𝑥))
1817equncomi 4120 . . . . . . 7 ( 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 dom (𝑅𝑟𝑥)) = ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)))
1913, 14, 183eqtri 2769 . . . . . 6 dom 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) = ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)))
20 rniun 6105 . . . . . . 7 ran 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) = 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)ran (𝑅𝑟𝑥)
21 iunxun 5059 . . . . . . 7 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)ran (𝑅𝑟𝑥) = ( 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))ran (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))
22 iunxun 5059 . . . . . . . 8 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))ran (𝑅𝑟𝑥) = ( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))
2322uneq1i 4124 . . . . . . 7 ( 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))ran (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)) = (( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))
2420, 21, 233eqtri 2769 . . . . . 6 ran 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) = (( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))
2519, 24uneq12i 4126 . . . . 5 (dom 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∪ ran 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥)) = (( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥))) ∪ (( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)))
26 uncom 4118 . . . . . . 7 ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥))) = (( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)) ∪ 𝑥𝑍 dom (𝑅𝑟𝑥))
2726uneq1i 4124 . . . . . 6 (( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥))) ∪ (( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))) = ((( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)) ∪ 𝑥𝑍 dom (𝑅𝑟𝑥)) ∪ (( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)))
28 un4 4134 . . . . . 6 ((( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)) ∪ 𝑥𝑍 dom (𝑅𝑟𝑥)) ∪ (( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))) = ((( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) ∪ ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)))
2927, 28eqtri 2765 . . . . 5 (( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥))) ∪ (( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))) = ((( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) ∪ ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)))
30 uncom 4118 . . . . . . . 8 ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)) = ( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥))
3130uneq1i 4124 . . . . . . 7 (( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)))
32 un4 4134 . . . . . . 7 (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)))
3331, 32eqtri 2765 . . . . . 6 (( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)))
3433uneq1i 4124 . . . . 5 ((( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) ∪ ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))) = ((( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) ∪ ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)))
3525, 29, 343eqtri 2769 . . . 4 (dom 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∪ ran 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥)) = ((( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) ∪ ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)))
36 df-ne 2945 . . . . . . . . . 10 (({0, 1} ∩ 𝑍) ≠ ∅ ↔ ¬ ({0, 1} ∩ 𝑍) = ∅)
37 incom 4166 . . . . . . . . . . . . . . 15 ({0, 1} ∩ 𝑍) = (𝑍 ∩ {0, 1})
381ineq2i 4174 . . . . . . . . . . . . . . 15 (𝑍 ∩ {0, 1}) = (𝑍 ∩ ({0} ∪ {1}))
39 indi 4238 . . . . . . . . . . . . . . 15 (𝑍 ∩ ({0} ∪ {1})) = ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1}))
4037, 38, 393eqtri 2769 . . . . . . . . . . . . . 14 ({0, 1} ∩ 𝑍) = ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1}))
4140eqeq1i 2742 . . . . . . . . . . . . 13 (({0, 1} ∩ 𝑍) = ∅ ↔ ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1})) = ∅)
42 un00 4407 . . . . . . . . . . . . 13 (((𝑍 ∩ {0}) = ∅ ∧ (𝑍 ∩ {1}) = ∅) ↔ ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1})) = ∅)
43 anor 982 . . . . . . . . . . . . 13 (((𝑍 ∩ {0}) = ∅ ∧ (𝑍 ∩ {1}) = ∅) ↔ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
4441, 42, 433bitr2i 299 . . . . . . . . . . . 12 (({0, 1} ∩ 𝑍) = ∅ ↔ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
4544notbii 320 . . . . . . . . . . 11 (¬ ({0, 1} ∩ 𝑍) = ∅ ↔ ¬ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
46 notnotb 315 . . . . . . . . . . 11 ((¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅) ↔ ¬ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
47 disjsn 4677 . . . . . . . . . . . . . 14 ((𝑍 ∩ {0}) = ∅ ↔ ¬ 0 ∈ 𝑍)
4847notbii 320 . . . . . . . . . . . . 13 (¬ (𝑍 ∩ {0}) = ∅ ↔ ¬ ¬ 0 ∈ 𝑍)
49 notnotb 315 . . . . . . . . . . . . 13 (0 ∈ 𝑍 ↔ ¬ ¬ 0 ∈ 𝑍)
5048, 49bitr4i 278 . . . . . . . . . . . 12 (¬ (𝑍 ∩ {0}) = ∅ ↔ 0 ∈ 𝑍)
51 disjsn 4677 . . . . . . . . . . . . . 14 ((𝑍 ∩ {1}) = ∅ ↔ ¬ 1 ∈ 𝑍)
5251notbii 320 . . . . . . . . . . . . 13 (¬ (𝑍 ∩ {1}) = ∅ ↔ ¬ ¬ 1 ∈ 𝑍)
53 notnotb 315 . . . . . . . . . . . . 13 (1 ∈ 𝑍 ↔ ¬ ¬ 1 ∈ 𝑍)
5452, 53bitr4i 278 . . . . . . . . . . . 12 (¬ (𝑍 ∩ {1}) = ∅ ↔ 1 ∈ 𝑍)
5550, 54orbi12i 914 . . . . . . . . . . 11 ((¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅) ↔ (0 ∈ 𝑍 ∨ 1 ∈ 𝑍))
5645, 46, 553bitr2i 299 . . . . . . . . . 10 (¬ ({0, 1} ∩ 𝑍) = ∅ ↔ (0 ∈ 𝑍 ∨ 1 ∈ 𝑍))
5736, 56sylbb 218 . . . . . . . . 9 (({0, 1} ∩ 𝑍) ≠ ∅ → (0 ∈ 𝑍 ∨ 1 ∈ 𝑍))
58 simpl 484 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 0 ∈ 𝑍)
5958snssd 4774 . . . . . . . . . . . . . . . . . . 19 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → {0} ⊆ 𝑍)
60 df-ss 3932 . . . . . . . . . . . . . . . . . . 19 ({0} ⊆ 𝑍 ↔ ({0} ∩ 𝑍) = {0})
6159, 60sylib 217 . . . . . . . . . . . . . . . . . 18 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({0} ∩ 𝑍) = {0})
6261iuneq1d 4986 . . . . . . . . . . . . . . . . 17 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) = 𝑥 ∈ {0}dom (𝑅𝑟𝑥))
63 c0ex 11156 . . . . . . . . . . . . . . . . . 18 0 ∈ V
64 oveq2 7370 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 0 → (𝑅𝑟𝑥) = (𝑅𝑟0))
6564dmeqd 5866 . . . . . . . . . . . . . . . . . 18 (𝑥 = 0 → dom (𝑅𝑟𝑥) = dom (𝑅𝑟0))
6663, 65iunxsn 5056 . . . . . . . . . . . . . . . . 17 𝑥 ∈ {0}dom (𝑅𝑟𝑥) = dom (𝑅𝑟0)
6762, 66eqtrdi 2793 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) = dom (𝑅𝑟0))
68 relexp0g 14914 . . . . . . . . . . . . . . . . . . 19 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
6968ad2antll 728 . . . . . . . . . . . . . . . . . 18 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
7069dmeqd 5866 . . . . . . . . . . . . . . . . 17 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → dom (𝑅𝑟0) = dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
71 dmresi 6010 . . . . . . . . . . . . . . . . 17 dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
7270, 71eqtrdi 2793 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → dom (𝑅𝑟0) = (dom 𝑅 ∪ ran 𝑅))
7367, 72eqtrd 2777 . . . . . . . . . . . . . . 15 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) = (dom 𝑅 ∪ ran 𝑅))
7461iuneq1d 4986 . . . . . . . . . . . . . . . . 17 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) = 𝑥 ∈ {0}ran (𝑅𝑟𝑥))
7564rneqd 5898 . . . . . . . . . . . . . . . . . 18 (𝑥 = 0 → ran (𝑅𝑟𝑥) = ran (𝑅𝑟0))
7663, 75iunxsn 5056 . . . . . . . . . . . . . . . . 17 𝑥 ∈ {0}ran (𝑅𝑟𝑥) = ran (𝑅𝑟0)
7774, 76eqtrdi 2793 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) = ran (𝑅𝑟0))
7869rneqd 5898 . . . . . . . . . . . . . . . . 17 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ran (𝑅𝑟0) = ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
79 rnresi 6032 . . . . . . . . . . . . . . . . 17 ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
8078, 79eqtrdi 2793 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ran (𝑅𝑟0) = (dom 𝑅 ∪ ran 𝑅))
8177, 80eqtrd 2777 . . . . . . . . . . . . . . 15 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) = (dom 𝑅 ∪ ran 𝑅))
8273, 81uneq12d 4129 . . . . . . . . . . . . . 14 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) = ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅)))
83 unidm 4117 . . . . . . . . . . . . . 14 ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
8482, 83eqtrdi 2793 . . . . . . . . . . . . 13 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) = (dom 𝑅 ∪ ran 𝑅))
8584uneq1d 4127 . . . . . . . . . . . 12 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = ((dom 𝑅 ∪ ran 𝑅) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))))
86 relexpdmg 14934 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℕ0𝑅𝑉) → dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
8786expcom 415 . . . . . . . . . . . . . . . . . 18 (𝑅𝑉 → (𝑥 ∈ ℕ0 → dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
8887ralrimiv 3143 . . . . . . . . . . . . . . . . 17 (𝑅𝑉 → ∀𝑥 ∈ ℕ0 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
8988ad2antll 728 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ∀𝑥 ∈ ℕ0 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
90 olc 867 . . . . . . . . . . . . . . . . . . . . 21 (𝑍 ⊆ ℕ0 → ({1} ⊆ ℕ0𝑍 ⊆ ℕ0))
9190ad2antrl 727 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({1} ⊆ ℕ0𝑍 ⊆ ℕ0))
92 inss 4203 . . . . . . . . . . . . . . . . . . . 20 (({1} ⊆ ℕ0𝑍 ⊆ ℕ0) → ({1} ∩ 𝑍) ⊆ ℕ0)
9391, 92syl 17 . . . . . . . . . . . . . . . . . . 19 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({1} ∩ 𝑍) ⊆ ℕ0)
9493sseld 3948 . . . . . . . . . . . . . . . . . 18 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (𝑥 ∈ ({1} ∩ 𝑍) → 𝑥 ∈ ℕ0))
9594imim1d 82 . . . . . . . . . . . . . . . . 17 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ((𝑥 ∈ ℕ0 → dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)) → (𝑥 ∈ ({1} ∩ 𝑍) → dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))))
9695ralimdv2 3161 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (∀𝑥 ∈ ℕ0 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) → ∀𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
9789, 96mpd 15 . . . . . . . . . . . . . . 15 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ∀𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
98 iunss 5010 . . . . . . . . . . . . . . 15 ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
9997, 98sylibr 233 . . . . . . . . . . . . . 14 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
100 relexprng 14938 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℕ0𝑅𝑉) → ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
101100expcom 415 . . . . . . . . . . . . . . . . . 18 (𝑅𝑉 → (𝑥 ∈ ℕ0 → ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
102101ralrimiv 3143 . . . . . . . . . . . . . . . . 17 (𝑅𝑉 → ∀𝑥 ∈ ℕ0 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
103102ad2antll 728 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ∀𝑥 ∈ ℕ0 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
10494imim1d 82 . . . . . . . . . . . . . . . . 17 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ((𝑥 ∈ ℕ0 → ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)) → (𝑥 ∈ ({1} ∩ 𝑍) → ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))))
105104ralimdv2 3161 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (∀𝑥 ∈ ℕ0 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) → ∀𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
106103, 105mpd 15 . . . . . . . . . . . . . . 15 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ∀𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
107 iunss 5010 . . . . . . . . . . . . . . 15 ( 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
108106, 107sylibr 233 . . . . . . . . . . . . . 14 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
10999, 108unssd 4151 . . . . . . . . . . . . 13 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
110 ssequn2 4148 . . . . . . . . . . . . 13 (( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ((dom 𝑅 ∪ ran 𝑅) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
111109, 110sylib 217 . . . . . . . . . . . 12 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ((dom 𝑅 ∪ ran 𝑅) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
11285, 111eqtrd 2777 . . . . . . . . . . 11 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
113112ex 414 . . . . . . . . . 10 (0 ∈ 𝑍 → ((𝑍 ⊆ ℕ0𝑅𝑉) → (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅)))
114 simpl 484 . . . . . . . . . . . . . . . . . . 19 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 1 ∈ 𝑍)
115114snssd 4774 . . . . . . . . . . . . . . . . . 18 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → {1} ⊆ 𝑍)
116 df-ss 3932 . . . . . . . . . . . . . . . . . 18 ({1} ⊆ 𝑍 ↔ ({1} ∩ 𝑍) = {1})
117115, 116sylib 217 . . . . . . . . . . . . . . . . 17 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({1} ∩ 𝑍) = {1})
118117iuneq1d 4986 . . . . . . . . . . . . . . . 16 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) = 𝑥 ∈ {1}dom (𝑅𝑟𝑥))
119 1ex 11158 . . . . . . . . . . . . . . . . 17 1 ∈ V
120 oveq2 7370 . . . . . . . . . . . . . . . . . 18 (𝑥 = 1 → (𝑅𝑟𝑥) = (𝑅𝑟1))
121120dmeqd 5866 . . . . . . . . . . . . . . . . 17 (𝑥 = 1 → dom (𝑅𝑟𝑥) = dom (𝑅𝑟1))
122119, 121iunxsn 5056 . . . . . . . . . . . . . . . 16 𝑥 ∈ {1}dom (𝑅𝑟𝑥) = dom (𝑅𝑟1)
123118, 122eqtrdi 2793 . . . . . . . . . . . . . . 15 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) = dom (𝑅𝑟1))
124 relexp1g 14918 . . . . . . . . . . . . . . . . 17 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
125124ad2antll 728 . . . . . . . . . . . . . . . 16 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (𝑅𝑟1) = 𝑅)
126125dmeqd 5866 . . . . . . . . . . . . . . 15 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → dom (𝑅𝑟1) = dom 𝑅)
127123, 126eqtrd 2777 . . . . . . . . . . . . . 14 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) = dom 𝑅)
128117iuneq1d 4986 . . . . . . . . . . . . . . . 16 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) = 𝑥 ∈ {1}ran (𝑅𝑟𝑥))
129120rneqd 5898 . . . . . . . . . . . . . . . . 17 (𝑥 = 1 → ran (𝑅𝑟𝑥) = ran (𝑅𝑟1))
130119, 129iunxsn 5056 . . . . . . . . . . . . . . . 16 𝑥 ∈ {1}ran (𝑅𝑟𝑥) = ran (𝑅𝑟1)
131128, 130eqtrdi 2793 . . . . . . . . . . . . . . 15 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) = ran (𝑅𝑟1))
132125rneqd 5898 . . . . . . . . . . . . . . 15 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ran (𝑅𝑟1) = ran 𝑅)
133131, 132eqtrd 2777 . . . . . . . . . . . . . 14 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) = ran 𝑅)
134127, 133uneq12d 4129 . . . . . . . . . . . . 13 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) = (dom 𝑅 ∪ ran 𝑅))
135134uneq2d 4128 . . . . . . . . . . . 12 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ (dom 𝑅 ∪ ran 𝑅)))
13688ad2antll 728 . . . . . . . . . . . . . . . 16 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ∀𝑥 ∈ ℕ0 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
137 olc 867 . . . . . . . . . . . . . . . . . . . . 21 (𝑍 ⊆ ℕ0 → ({0} ⊆ ℕ0𝑍 ⊆ ℕ0))
138137ad2antrl 727 . . . . . . . . . . . . . . . . . . . 20 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({0} ⊆ ℕ0𝑍 ⊆ ℕ0))
139 inss 4203 . . . . . . . . . . . . . . . . . . . 20 (({0} ⊆ ℕ0𝑍 ⊆ ℕ0) → ({0} ∩ 𝑍) ⊆ ℕ0)
140138, 139syl 17 . . . . . . . . . . . . . . . . . . 19 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({0} ∩ 𝑍) ⊆ ℕ0)
141140sseld 3948 . . . . . . . . . . . . . . . . . 18 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (𝑥 ∈ ({0} ∩ 𝑍) → 𝑥 ∈ ℕ0))
142141imim1d 82 . . . . . . . . . . . . . . . . 17 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ((𝑥 ∈ ℕ0 → dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)) → (𝑥 ∈ ({0} ∩ 𝑍) → dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))))
143142ralimdv2 3161 . . . . . . . . . . . . . . . 16 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (∀𝑥 ∈ ℕ0 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) → ∀𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
144136, 143mpd 15 . . . . . . . . . . . . . . 15 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ∀𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
145 iunss 5010 . . . . . . . . . . . . . . 15 ( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
146144, 145sylibr 233 . . . . . . . . . . . . . 14 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
147102ad2antll 728 . . . . . . . . . . . . . . . 16 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ∀𝑥 ∈ ℕ0 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
148141imim1d 82 . . . . . . . . . . . . . . . . 17 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ((𝑥 ∈ ℕ0 → ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)) → (𝑥 ∈ ({0} ∩ 𝑍) → ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))))
149148ralimdv2 3161 . . . . . . . . . . . . . . . 16 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (∀𝑥 ∈ ℕ0 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) → ∀𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
150147, 149mpd 15 . . . . . . . . . . . . . . 15 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ∀𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
151 iunss 5010 . . . . . . . . . . . . . . 15 ( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
152150, 151sylibr 233 . . . . . . . . . . . . . 14 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
153146, 152unssd 4151 . . . . . . . . . . . . 13 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
154 ssequn1 4145 . . . . . . . . . . . . 13 (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅))
155153, 154sylib 217 . . . . . . . . . . . 12 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅))
156135, 155eqtrd 2777 . . . . . . . . . . 11 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
157156ex 414 . . . . . . . . . 10 (1 ∈ 𝑍 → ((𝑍 ⊆ ℕ0𝑅𝑉) → (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅)))
158113, 157jaoi 856 . . . . . . . . 9 ((0 ∈ 𝑍 ∨ 1 ∈ 𝑍) → ((𝑍 ⊆ ℕ0𝑅𝑉) → (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅)))
15957, 158syl 17 . . . . . . . 8 (({0, 1} ∩ 𝑍) ≠ ∅ → ((𝑍 ⊆ ℕ0𝑅𝑉) → (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅)))
1601593impib 1117 . . . . . . 7 ((({0, 1} ∩ 𝑍) ≠ ∅ ∧ 𝑍 ⊆ ℕ0𝑅𝑉) → (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
1611603com13 1125 . . . . . 6 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
162161uneq1d 4127 . . . . 5 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ((( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) ∪ ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))) = ((dom 𝑅 ∪ ran 𝑅) ∪ ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))))
16388adantr 482 . . . . . . . . . 10 ((𝑅𝑉𝑍 ⊆ ℕ0) → ∀𝑥 ∈ ℕ0 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
164 ssel 3942 . . . . . . . . . . . . 13 (𝑍 ⊆ ℕ0 → (𝑥𝑍𝑥 ∈ ℕ0))
165164adantl 483 . . . . . . . . . . . 12 ((𝑅𝑉𝑍 ⊆ ℕ0) → (𝑥𝑍𝑥 ∈ ℕ0))
166165imim1d 82 . . . . . . . . . . 11 ((𝑅𝑉𝑍 ⊆ ℕ0) → ((𝑥 ∈ ℕ0 → dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)) → (𝑥𝑍 → dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))))
167166ralimdv2 3161 . . . . . . . . . 10 ((𝑅𝑉𝑍 ⊆ ℕ0) → (∀𝑥 ∈ ℕ0 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) → ∀𝑥𝑍 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
168163, 167mpd 15 . . . . . . . . 9 ((𝑅𝑉𝑍 ⊆ ℕ0) → ∀𝑥𝑍 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
169 iunss 5010 . . . . . . . . 9 ( 𝑥𝑍 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥𝑍 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
170168, 169sylibr 233 . . . . . . . 8 ((𝑅𝑉𝑍 ⊆ ℕ0) → 𝑥𝑍 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
171102adantr 482 . . . . . . . . . 10 ((𝑅𝑉𝑍 ⊆ ℕ0) → ∀𝑥 ∈ ℕ0 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
172165imim1d 82 . . . . . . . . . . 11 ((𝑅𝑉𝑍 ⊆ ℕ0) → ((𝑥 ∈ ℕ0 → ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)) → (𝑥𝑍 → ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))))
173172ralimdv2 3161 . . . . . . . . . 10 ((𝑅𝑉𝑍 ⊆ ℕ0) → (∀𝑥 ∈ ℕ0 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) → ∀𝑥𝑍 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
174171, 173mpd 15 . . . . . . . . 9 ((𝑅𝑉𝑍 ⊆ ℕ0) → ∀𝑥𝑍 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
175 iunss 5010 . . . . . . . . 9 ( 𝑥𝑍 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥𝑍 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
176174, 175sylibr 233 . . . . . . . 8 ((𝑅𝑉𝑍 ⊆ ℕ0) → 𝑥𝑍 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
177170, 176unssd 4151 . . . . . . 7 ((𝑅𝑉𝑍 ⊆ ℕ0) → ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
1781773adant3 1133 . . . . . 6 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
179 ssequn2 4148 . . . . . 6 (( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ((dom 𝑅 ∪ ran 𝑅) ∪ ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
180178, 179sylib 217 . . . . 5 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ((dom 𝑅 ∪ ran 𝑅) ∪ ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
181162, 180eqtrd 2777 . . . 4 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ((( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) ∪ ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
18235, 181eqtrid 2789 . . 3 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → (dom 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∪ ran 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥)) = (dom 𝑅 ∪ ran 𝑅))
183 nn0ex 12426 . . . . . . 7 0 ∈ V
184183ssex 5283 . . . . . 6 (𝑍 ⊆ ℕ0𝑍 ∈ V)
185 inex2g 5282 . . . . . . . . 9 (𝑍 ∈ V → ({0} ∩ 𝑍) ∈ V)
186 inex2g 5282 . . . . . . . . 9 (𝑍 ∈ V → ({1} ∩ 𝑍) ∈ V)
187185, 186unexd 7693 . . . . . . . 8 (𝑍 ∈ V → (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∈ V)
188 unexg 7688 . . . . . . . 8 (((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∈ V ∧ 𝑍 ∈ V) → ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) ∈ V)
189187, 188mpancom 687 . . . . . . 7 (𝑍 ∈ V → ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) ∈ V)
190 ovex 7395 . . . . . . . 8 (𝑅𝑟𝑥) ∈ V
191190rgenw 3069 . . . . . . 7 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∈ V
192 iunexg 7901 . . . . . . 7 ((((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) ∈ V ∧ ∀𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∈ V) → 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∈ V)
193189, 191, 192sylancl 587 . . . . . 6 (𝑍 ∈ V → 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∈ V)
194184, 193syl 17 . . . . 5 (𝑍 ⊆ ℕ0 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∈ V)
1951943ad2ant2 1135 . . . 4 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∈ V)
196 simp1 1137 . . . 4 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → 𝑅𝑉)
197 relexp0eq 42047 . . . 4 (( 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∈ V ∧ 𝑅𝑉) → ((dom 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∪ ran 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥)) = (dom 𝑅 ∪ ran 𝑅) ↔ ( 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥)↑𝑟0) = (𝑅𝑟0)))
198195, 196, 197syl2anc 585 . . 3 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ((dom 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∪ ran 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥)) = (dom 𝑅 ∪ ran 𝑅) ↔ ( 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥)↑𝑟0) = (𝑅𝑟0)))
199182, 198mpbid 231 . 2 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ( 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥)↑𝑟0) = (𝑅𝑟0))
20012, 199eqtrid 2789 1 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ( 𝑥𝑍 (𝑅𝑟𝑥)↑𝑟0) = (𝑅𝑟0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wne 2944  wral 3065  Vcvv 3448  cun 3913  cin 3914  wss 3915  c0 4287  {csn 4591  {cpr 4593   ciun 4959   I cid 5535  dom cdm 5638  ran crn 5639  cres 5640  (class class class)co 7362  0cc0 11058  1c1 11059  0cn0 12420  𝑟crelexp 14911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-n0 12421  df-z 12507  df-uz 12771  df-seq 13914  df-relexp 14912
This theorem is referenced by:  corclrcl  42053  corcltrcl  42085
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