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Theorem iunrelexp0 43196
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 4627 . . . . . . 7 {0, 1} = ({0} ∪ {1})
21ineq1i 4202 . . . . . 6 ({0, 1} ∩ 𝑍) = (({0} ∪ {1}) ∩ 𝑍)
3 indir 4270 . . . . . 6 (({0} ∪ {1}) ∩ 𝑍) = (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))
42, 3eqtr2i 2754 . . . . 5 (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) = ({0, 1} ∩ 𝑍)
54uneq1i 4152 . . . 4 ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) = (({0, 1} ∩ 𝑍) ∪ 𝑍)
6 inss2 4224 . . . . 5 ({0, 1} ∩ 𝑍) ⊆ 𝑍
7 ssequn1 4174 . . . . 5 (({0, 1} ∩ 𝑍) ⊆ 𝑍 ↔ (({0, 1} ∩ 𝑍) ∪ 𝑍) = 𝑍)
86, 7mpbi 229 . . . 4 (({0, 1} ∩ 𝑍) ∪ 𝑍) = 𝑍
95, 8eqtr2i 2754 . . 3 𝑍 = ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)
10 iuneq1 5007 . . . 4 (𝑍 = ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) → 𝑥𝑍 (𝑅𝑟𝑥) = 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥))
1110oveq1d 7430 . . 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 5910 . . . . . . 7 dom 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) = 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)dom (𝑅𝑟𝑥)
14 iunxun 5092 . . . . . . 7 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)dom (𝑅𝑟𝑥) = ( 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 dom (𝑅𝑟𝑥))
15 iunxun 5092 . . . . . . . . . 10 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅𝑟𝑥) = ( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥))
1615equncomi 4148 . . . . . . . . 9 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅𝑟𝑥) = ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥))
1716uneq1i 4152 . . . . . . . 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 4148 . . . . . . 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 2757 . . . . . 6 dom 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) = ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)))
20 rniun 6147 . . . . . . 7 ran 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) = 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)ran (𝑅𝑟𝑥)
21 iunxun 5092 . . . . . . 7 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)ran (𝑅𝑟𝑥) = ( 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))ran (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))
22 iunxun 5092 . . . . . . . 8 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))ran (𝑅𝑟𝑥) = ( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))
2322uneq1i 4152 . . . . . . 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 2757 . . . . . 6 ran 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) = (( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))
2519, 24uneq12i 4154 . . . . 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 4146 . . . . . . 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 4152 . . . . . 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 4163 . . . . . 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 2753 . . . . 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 4146 . . . . . . . 8 ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)) = ( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥))
3130uneq1i 4152 . . . . . . 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 4163 . . . . . . 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 2753 . . . . . 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 4152 . . . . 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 2757 . . . 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 2931 . . . . . . . . . 10 (({0, 1} ∩ 𝑍) ≠ ∅ ↔ ¬ ({0, 1} ∩ 𝑍) = ∅)
37 incom 4195 . . . . . . . . . . . . . . 15 ({0, 1} ∩ 𝑍) = (𝑍 ∩ {0, 1})
381ineq2i 4203 . . . . . . . . . . . . . . 15 (𝑍 ∩ {0, 1}) = (𝑍 ∩ ({0} ∪ {1}))
39 indi 4268 . . . . . . . . . . . . . . 15 (𝑍 ∩ ({0} ∪ {1})) = ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1}))
4037, 38, 393eqtri 2757 . . . . . . . . . . . . . 14 ({0, 1} ∩ 𝑍) = ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1}))
4140eqeq1i 2730 . . . . . . . . . . . . 13 (({0, 1} ∩ 𝑍) = ∅ ↔ ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1})) = ∅)
42 un00 4438 . . . . . . . . . . . . 13 (((𝑍 ∩ {0}) = ∅ ∧ (𝑍 ∩ {1}) = ∅) ↔ ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1})) = ∅)
43 anor 980 . . . . . . . . . . . . 13 (((𝑍 ∩ {0}) = ∅ ∧ (𝑍 ∩ {1}) = ∅) ↔ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
4441, 42, 433bitr2i 298 . . . . . . . . . . . 12 (({0, 1} ∩ 𝑍) = ∅ ↔ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
4544notbii 319 . . . . . . . . . . 11 (¬ ({0, 1} ∩ 𝑍) = ∅ ↔ ¬ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
46 notnotb 314 . . . . . . . . . . 11 ((¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅) ↔ ¬ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
47 disjsn 4711 . . . . . . . . . . . . . 14 ((𝑍 ∩ {0}) = ∅ ↔ ¬ 0 ∈ 𝑍)
4847notbii 319 . . . . . . . . . . . . 13 (¬ (𝑍 ∩ {0}) = ∅ ↔ ¬ ¬ 0 ∈ 𝑍)
49 notnotb 314 . . . . . . . . . . . . 13 (0 ∈ 𝑍 ↔ ¬ ¬ 0 ∈ 𝑍)
5048, 49bitr4i 277 . . . . . . . . . . . 12 (¬ (𝑍 ∩ {0}) = ∅ ↔ 0 ∈ 𝑍)
51 disjsn 4711 . . . . . . . . . . . . . 14 ((𝑍 ∩ {1}) = ∅ ↔ ¬ 1 ∈ 𝑍)
5251notbii 319 . . . . . . . . . . . . 13 (¬ (𝑍 ∩ {1}) = ∅ ↔ ¬ ¬ 1 ∈ 𝑍)
53 notnotb 314 . . . . . . . . . . . . 13 (1 ∈ 𝑍 ↔ ¬ ¬ 1 ∈ 𝑍)
5452, 53bitr4i 277 . . . . . . . . . . . 12 (¬ (𝑍 ∩ {1}) = ∅ ↔ 1 ∈ 𝑍)
5550, 54orbi12i 912 . . . . . . . . . . 11 ((¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅) ↔ (0 ∈ 𝑍 ∨ 1 ∈ 𝑍))
5645, 46, 553bitr2i 298 . . . . . . . . . 10 (¬ ({0, 1} ∩ 𝑍) = ∅ ↔ (0 ∈ 𝑍 ∨ 1 ∈ 𝑍))
5736, 56sylbb 218 . . . . . . . . 9 (({0, 1} ∩ 𝑍) ≠ ∅ → (0 ∈ 𝑍 ∨ 1 ∈ 𝑍))
58 simpl 481 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 0 ∈ 𝑍)
5958snssd 4808 . . . . . . . . . . . . . . . . . . 19 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → {0} ⊆ 𝑍)
60 dfss2 3958 . . . . . . . . . . . . . . . . . . 19 ({0} ⊆ 𝑍 ↔ ({0} ∩ 𝑍) = {0})
6159, 60sylib 217 . . . . . . . . . . . . . . . . . 18 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({0} ∩ 𝑍) = {0})
6261iuneq1d 5018 . . . . . . . . . . . . . . . . 17 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) = 𝑥 ∈ {0}dom (𝑅𝑟𝑥))
63 c0ex 11236 . . . . . . . . . . . . . . . . . 18 0 ∈ V
64 oveq2 7423 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 0 → (𝑅𝑟𝑥) = (𝑅𝑟0))
6564dmeqd 5902 . . . . . . . . . . . . . . . . . 18 (𝑥 = 0 → dom (𝑅𝑟𝑥) = dom (𝑅𝑟0))
6663, 65iunxsn 5089 . . . . . . . . . . . . . . . . 17 𝑥 ∈ {0}dom (𝑅𝑟𝑥) = dom (𝑅𝑟0)
6762, 66eqtrdi 2781 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) = dom (𝑅𝑟0))
68 relexp0g 14999 . . . . . . . . . . . . . . . . . . 19 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
6968ad2antll 727 . . . . . . . . . . . . . . . . . 18 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
7069dmeqd 5902 . . . . . . . . . . . . . . . . 17 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → dom (𝑅𝑟0) = dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
71 dmresi 6050 . . . . . . . . . . . . . . . . 17 dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
7270, 71eqtrdi 2781 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → dom (𝑅𝑟0) = (dom 𝑅 ∪ ran 𝑅))
7367, 72eqtrd 2765 . . . . . . . . . . . . . . 15 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) = (dom 𝑅 ∪ ran 𝑅))
7461iuneq1d 5018 . . . . . . . . . . . . . . . . 17 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) = 𝑥 ∈ {0}ran (𝑅𝑟𝑥))
7564rneqd 5934 . . . . . . . . . . . . . . . . . 18 (𝑥 = 0 → ran (𝑅𝑟𝑥) = ran (𝑅𝑟0))
7663, 75iunxsn 5089 . . . . . . . . . . . . . . . . 17 𝑥 ∈ {0}ran (𝑅𝑟𝑥) = ran (𝑅𝑟0)
7774, 76eqtrdi 2781 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) = ran (𝑅𝑟0))
7869rneqd 5934 . . . . . . . . . . . . . . . . 17 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ran (𝑅𝑟0) = ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
79 rnresi 6073 . . . . . . . . . . . . . . . . 17 ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
8078, 79eqtrdi 2781 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ran (𝑅𝑟0) = (dom 𝑅 ∪ ran 𝑅))
8177, 80eqtrd 2765 . . . . . . . . . . . . . . 15 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) = (dom 𝑅 ∪ ran 𝑅))
8273, 81uneq12d 4157 . . . . . . . . . . . . . 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 4145 . . . . . . . . . . . . . 14 ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
8482, 83eqtrdi 2781 . . . . . . . . . . . . 13 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) = (dom 𝑅 ∪ ran 𝑅))
8584uneq1d 4155 . . . . . . . . . . . 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 15019 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℕ0𝑅𝑉) → dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
8786expcom 412 . . . . . . . . . . . . . . . . . 18 (𝑅𝑉 → (𝑥 ∈ ℕ0 → dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
8887ralrimiv 3135 . . . . . . . . . . . . . . . . 17 (𝑅𝑉 → ∀𝑥 ∈ ℕ0 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
8988ad2antll 727 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ∀𝑥 ∈ ℕ0 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
90 olc 866 . . . . . . . . . . . . . . . . . . . . 21 (𝑍 ⊆ ℕ0 → ({1} ⊆ ℕ0𝑍 ⊆ ℕ0))
9190ad2antrl 726 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({1} ⊆ ℕ0𝑍 ⊆ ℕ0))
92 inss 4233 . . . . . . . . . . . . . . . . . . . 20 (({1} ⊆ ℕ0𝑍 ⊆ ℕ0) → ({1} ∩ 𝑍) ⊆ ℕ0)
9391, 92syl 17 . . . . . . . . . . . . . . . . . . 19 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({1} ∩ 𝑍) ⊆ ℕ0)
9493sseld 3971 . . . . . . . . . . . . . . . . . 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 3153 . . . . . . . . . . . . . . . 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 5043 . . . . . . . . . . . . . . 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 15023 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℕ0𝑅𝑉) → ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
101100expcom 412 . . . . . . . . . . . . . . . . . 18 (𝑅𝑉 → (𝑥 ∈ ℕ0 → ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
102101ralrimiv 3135 . . . . . . . . . . . . . . . . 17 (𝑅𝑉 → ∀𝑥 ∈ ℕ0 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
103102ad2antll 727 . . . . . . . . . . . . . . . 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 3153 . . . . . . . . . . . . . . . 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 5043 . . . . . . . . . . . . . . 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 4180 . . . . . . . . . . . . 13 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
110 ssequn2 4177 . . . . . . . . . . . . 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 2765 . . . . . . . . . . 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 411 . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . 19 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 1 ∈ 𝑍)
115114snssd 4808 . . . . . . . . . . . . . . . . . 18 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → {1} ⊆ 𝑍)
116 dfss2 3958 . . . . . . . . . . . . . . . . . 18 ({1} ⊆ 𝑍 ↔ ({1} ∩ 𝑍) = {1})
117115, 116sylib 217 . . . . . . . . . . . . . . . . 17 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({1} ∩ 𝑍) = {1})
118117iuneq1d 5018 . . . . . . . . . . . . . . . 16 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) = 𝑥 ∈ {1}dom (𝑅𝑟𝑥))
119 1ex 11238 . . . . . . . . . . . . . . . . 17 1 ∈ V
120 oveq2 7423 . . . . . . . . . . . . . . . . . 18 (𝑥 = 1 → (𝑅𝑟𝑥) = (𝑅𝑟1))
121120dmeqd 5902 . . . . . . . . . . . . . . . . 17 (𝑥 = 1 → dom (𝑅𝑟𝑥) = dom (𝑅𝑟1))
122119, 121iunxsn 5089 . . . . . . . . . . . . . . . 16 𝑥 ∈ {1}dom (𝑅𝑟𝑥) = dom (𝑅𝑟1)
123118, 122eqtrdi 2781 . . . . . . . . . . . . . . 15 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) = dom (𝑅𝑟1))
124 relexp1g 15003 . . . . . . . . . . . . . . . . 17 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
125124ad2antll 727 . . . . . . . . . . . . . . . 16 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (𝑅𝑟1) = 𝑅)
126125dmeqd 5902 . . . . . . . . . . . . . . 15 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → dom (𝑅𝑟1) = dom 𝑅)
127123, 126eqtrd 2765 . . . . . . . . . . . . . 14 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) = dom 𝑅)
128117iuneq1d 5018 . . . . . . . . . . . . . . . 16 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) = 𝑥 ∈ {1}ran (𝑅𝑟𝑥))
129120rneqd 5934 . . . . . . . . . . . . . . . . 17 (𝑥 = 1 → ran (𝑅𝑟𝑥) = ran (𝑅𝑟1))
130119, 129iunxsn 5089 . . . . . . . . . . . . . . . 16 𝑥 ∈ {1}ran (𝑅𝑟𝑥) = ran (𝑅𝑟1)
131128, 130eqtrdi 2781 . . . . . . . . . . . . . . 15 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) = ran (𝑅𝑟1))
132125rneqd 5934 . . . . . . . . . . . . . . 15 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ran (𝑅𝑟1) = ran 𝑅)
133131, 132eqtrd 2765 . . . . . . . . . . . . . 14 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) = ran 𝑅)
134127, 133uneq12d 4157 . . . . . . . . . . . . 13 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) = (dom 𝑅 ∪ ran 𝑅))
135134uneq2d 4156 . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . 16 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ∀𝑥 ∈ ℕ0 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
137 olc 866 . . . . . . . . . . . . . . . . . . . . 21 (𝑍 ⊆ ℕ0 → ({0} ⊆ ℕ0𝑍 ⊆ ℕ0))
138137ad2antrl 726 . . . . . . . . . . . . . . . . . . . 20 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({0} ⊆ ℕ0𝑍 ⊆ ℕ0))
139 inss 4233 . . . . . . . . . . . . . . . . . . . 20 (({0} ⊆ ℕ0𝑍 ⊆ ℕ0) → ({0} ∩ 𝑍) ⊆ ℕ0)
140138, 139syl 17 . . . . . . . . . . . . . . . . . . 19 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({0} ∩ 𝑍) ⊆ ℕ0)
141140sseld 3971 . . . . . . . . . . . . . . . . . 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 3153 . . . . . . . . . . . . . . . 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 5043 . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . 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 3153 . . . . . . . . . . . . . . . 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 5043 . . . . . . . . . . . . . . 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 4180 . . . . . . . . . . . . 13 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
154 ssequn1 4174 . . . . . . . . . . . . 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 2765 . . . . . . . . . . 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 411 . . . . . . . . . 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 855 . . . . . . . . 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 1113 . . . . . . 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 1121 . . . . . 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 4155 . . . . 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 479 . . . . . . . . . 10 ((𝑅𝑉𝑍 ⊆ ℕ0) → ∀𝑥 ∈ ℕ0 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
164 ssel 3966 . . . . . . . . . . . . 13 (𝑍 ⊆ ℕ0 → (𝑥𝑍𝑥 ∈ ℕ0))
165164adantl 480 . . . . . . . . . . . 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 3153 . . . . . . . . . 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 5043 . . . . . . . . 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 479 . . . . . . . . . 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 3153 . . . . . . . . . 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 5043 . . . . . . . . 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 4180 . . . . . . 7 ((𝑅𝑉𝑍 ⊆ ℕ0) → ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
1781773adant3 1129 . . . . . 6 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
179 ssequn2 4177 . . . . . 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 2765 . . . 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 2777 . . 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 12506 . . . . . . 7 0 ∈ V
184183ssex 5316 . . . . . 6 (𝑍 ⊆ ℕ0𝑍 ∈ V)
185 inex2g 5315 . . . . . . . . 9 (𝑍 ∈ V → ({0} ∩ 𝑍) ∈ V)
186 inex2g 5315 . . . . . . . . 9 (𝑍 ∈ V → ({1} ∩ 𝑍) ∈ V)
187185, 186unexd 7753 . . . . . . . 8 (𝑍 ∈ V → (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∈ V)
188 unexg 7748 . . . . . . . 8 (((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∈ V ∧ 𝑍 ∈ V) → ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) ∈ V)
189187, 188mpancom 686 . . . . . . 7 (𝑍 ∈ V → ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) ∈ V)
190 ovex 7448 . . . . . . . 8 (𝑅𝑟𝑥) ∈ V
191190rgenw 3055 . . . . . . 7 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∈ V
192 iunexg 7963 . . . . . . 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 584 . . . . . 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 1131 . . . 4 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∈ V)
196 simp1 1133 . . . 4 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → 𝑅𝑉)
197 relexp0eq 43195 . . . 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 582 . . 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 2777 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 394  wo 845  w3a 1084   = wceq 1533  wcel 2098  wne 2930  wral 3051  Vcvv 3463  cun 3938  cin 3939  wss 3940  c0 4318  {csn 4624  {cpr 4626   ciun 4991   I cid 5569  dom cdm 5672  ran crn 5673  cres 5674  (class class class)co 7415  0cc0 11136  1c1 11137  0cn0 12500  𝑟crelexp 14996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-er 8721  df-en 8961  df-dom 8962  df-sdom 8963  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-n0 12501  df-z 12587  df-uz 12851  df-seq 13997  df-relexp 14997
This theorem is referenced by:  corclrcl  43201  corcltrcl  43233
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