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Theorem iunrelexp0 39553
Description: Simplification of zeroth power of indexed union of powers of relations. (Contributed by RP, 19-Jun-2020.)
Assertion
Ref Expression
iunrelexp0 ((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 4481 . . . . . . 7 {0, 1} = ({0} ∪ {1})
21ineq1i 4111 . . . . . 6 ({0, 1} ∩ 𝑍) = (({0} ∪ {1}) ∩ 𝑍)
3 indir 4178 . . . . . 6 (({0} ∪ {1}) ∩ 𝑍) = (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))
42, 3eqtr2i 2822 . . . . 5 (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) = ({0, 1} ∩ 𝑍)
54uneq1i 4062 . . . 4 ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) = (({0, 1} ∩ 𝑍) ∪ 𝑍)
6 inss2 4132 . . . . 5 ({0, 1} ∩ 𝑍) ⊆ 𝑍
7 ssequn1 4083 . . . . 5 (({0, 1} ∩ 𝑍) ⊆ 𝑍 ↔ (({0, 1} ∩ 𝑍) ∪ 𝑍) = 𝑍)
86, 7mpbi 231 . . . 4 (({0, 1} ∩ 𝑍) ∪ 𝑍) = 𝑍
95, 8eqtr2i 2822 . . 3 𝑍 = ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)
10 iuneq1 4846 . . . 4 (𝑍 = ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) → 𝑥𝑍 (𝑅𝑟𝑥) = 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥))
1110oveq1d 7038 . . 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 5675 . . . . . . 7 dom 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) = 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)dom (𝑅𝑟𝑥)
14 iunxun 4921 . . . . . . 7 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)dom (𝑅𝑟𝑥) = ( 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 dom (𝑅𝑟𝑥))
15 iunxun 4921 . . . . . . . . . 10 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅𝑟𝑥) = ( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥))
1615equncomi 4058 . . . . . . . . 9 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅𝑟𝑥) = ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥))
1716uneq1i 4062 . . . . . . . 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 4058 . . . . . . 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 2825 . . . . . 6 dom 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) = ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)))
20 rniun 5889 . . . . . . 7 ran 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) = 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)ran (𝑅𝑟𝑥)
21 iunxun 4921 . . . . . . 7 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)ran (𝑅𝑟𝑥) = ( 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))ran (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))
22 iunxun 4921 . . . . . . . 8 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))ran (𝑅𝑟𝑥) = ( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))
2322uneq1i 4062 . . . . . . 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 2825 . . . . . 6 ran 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) = (( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))
2519, 24uneq12i 4064 . . . . 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 4056 . . . . . . 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 4062 . . . . . 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 4072 . . . . . 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 2821 . . . . 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 4056 . . . . . . . 8 ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)) = ( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥))
3130uneq1i 4062 . . . . . . 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 4072 . . . . . . 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 2821 . . . . . 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 4062 . . . . 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 2825 . . . 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 2987 . . . . . . . . . 10 (({0, 1} ∩ 𝑍) ≠ ∅ ↔ ¬ ({0, 1} ∩ 𝑍) = ∅)
37 incom 4105 . . . . . . . . . . . . . . 15 ({0, 1} ∩ 𝑍) = (𝑍 ∩ {0, 1})
381ineq2i 4112 . . . . . . . . . . . . . . 15 (𝑍 ∩ {0, 1}) = (𝑍 ∩ ({0} ∪ {1}))
39 indi 4176 . . . . . . . . . . . . . . 15 (𝑍 ∩ ({0} ∪ {1})) = ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1}))
4037, 38, 393eqtri 2825 . . . . . . . . . . . . . 14 ({0, 1} ∩ 𝑍) = ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1}))
4140eqeq1i 2802 . . . . . . . . . . . . 13 (({0, 1} ∩ 𝑍) = ∅ ↔ ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1})) = ∅)
42 un00 4314 . . . . . . . . . . . . 13 (((𝑍 ∩ {0}) = ∅ ∧ (𝑍 ∩ {1}) = ∅) ↔ ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1})) = ∅)
43 anor 977 . . . . . . . . . . . . 13 (((𝑍 ∩ {0}) = ∅ ∧ (𝑍 ∩ {1}) = ∅) ↔ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
4441, 42, 433bitr2i 300 . . . . . . . . . . . 12 (({0, 1} ∩ 𝑍) = ∅ ↔ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
4544notbii 321 . . . . . . . . . . 11 (¬ ({0, 1} ∩ 𝑍) = ∅ ↔ ¬ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
46 notnotb 316 . . . . . . . . . . 11 ((¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅) ↔ ¬ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
47 disjsn 4560 . . . . . . . . . . . . . 14 ((𝑍 ∩ {0}) = ∅ ↔ ¬ 0 ∈ 𝑍)
4847notbii 321 . . . . . . . . . . . . 13 (¬ (𝑍 ∩ {0}) = ∅ ↔ ¬ ¬ 0 ∈ 𝑍)
49 notnotb 316 . . . . . . . . . . . . 13 (0 ∈ 𝑍 ↔ ¬ ¬ 0 ∈ 𝑍)
5048, 49bitr4i 279 . . . . . . . . . . . 12 (¬ (𝑍 ∩ {0}) = ∅ ↔ 0 ∈ 𝑍)
51 disjsn 4560 . . . . . . . . . . . . . 14 ((𝑍 ∩ {1}) = ∅ ↔ ¬ 1 ∈ 𝑍)
5251notbii 321 . . . . . . . . . . . . 13 (¬ (𝑍 ∩ {1}) = ∅ ↔ ¬ ¬ 1 ∈ 𝑍)
53 notnotb 316 . . . . . . . . . . . . 13 (1 ∈ 𝑍 ↔ ¬ ¬ 1 ∈ 𝑍)
5452, 53bitr4i 279 . . . . . . . . . . . 12 (¬ (𝑍 ∩ {1}) = ∅ ↔ 1 ∈ 𝑍)
5550, 54orbi12i 909 . . . . . . . . . . 11 ((¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅) ↔ (0 ∈ 𝑍 ∨ 1 ∈ 𝑍))
5645, 46, 553bitr2i 300 . . . . . . . . . 10 (¬ ({0, 1} ∩ 𝑍) = ∅ ↔ (0 ∈ 𝑍 ∨ 1 ∈ 𝑍))
5736, 56sylbb 220 . . . . . . . . 9 (({0, 1} ∩ 𝑍) ≠ ∅ → (0 ∈ 𝑍 ∨ 1 ∈ 𝑍))
58 simpl 483 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 0 ∈ 𝑍)
5958snssd 4655 . . . . . . . . . . . . . . . . . . 19 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → {0} ⊆ 𝑍)
60 df-ss 3880 . . . . . . . . . . . . . . . . . . 19 ({0} ⊆ 𝑍 ↔ ({0} ∩ 𝑍) = {0})
6159, 60sylib 219 . . . . . . . . . . . . . . . . . 18 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({0} ∩ 𝑍) = {0})
6261iuneq1d 4857 . . . . . . . . . . . . . . . . 17 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) = 𝑥 ∈ {0}dom (𝑅𝑟𝑥))
63 c0ex 10488 . . . . . . . . . . . . . . . . . 18 0 ∈ V
64 oveq2 7031 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 0 → (𝑅𝑟𝑥) = (𝑅𝑟0))
6564dmeqd 5667 . . . . . . . . . . . . . . . . . 18 (𝑥 = 0 → dom (𝑅𝑟𝑥) = dom (𝑅𝑟0))
6663, 65iunxsn 4918 . . . . . . . . . . . . . . . . 17 𝑥 ∈ {0}dom (𝑅𝑟𝑥) = dom (𝑅𝑟0)
6762, 66syl6eq 2849 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) = dom (𝑅𝑟0))
68 relexp0g 14219 . . . . . . . . . . . . . . . . . . 19 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
6968ad2antll 725 . . . . . . . . . . . . . . . . . 18 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
7069dmeqd 5667 . . . . . . . . . . . . . . . . 17 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → dom (𝑅𝑟0) = dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
71 dmresi 5806 . . . . . . . . . . . . . . . . 17 dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
7270, 71syl6eq 2849 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → dom (𝑅𝑟0) = (dom 𝑅 ∪ ran 𝑅))
7367, 72eqtrd 2833 . . . . . . . . . . . . . . 15 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) = (dom 𝑅 ∪ ran 𝑅))
7461iuneq1d 4857 . . . . . . . . . . . . . . . . 17 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) = 𝑥 ∈ {0}ran (𝑅𝑟𝑥))
7564rneqd 5697 . . . . . . . . . . . . . . . . . 18 (𝑥 = 0 → ran (𝑅𝑟𝑥) = ran (𝑅𝑟0))
7663, 75iunxsn 4918 . . . . . . . . . . . . . . . . 17 𝑥 ∈ {0}ran (𝑅𝑟𝑥) = ran (𝑅𝑟0)
7774, 76syl6eq 2849 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) = ran (𝑅𝑟0))
7869rneqd 5697 . . . . . . . . . . . . . . . . 17 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ran (𝑅𝑟0) = ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
79 rnresi 5826 . . . . . . . . . . . . . . . . 17 ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
8078, 79syl6eq 2849 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ran (𝑅𝑟0) = (dom 𝑅 ∪ ran 𝑅))
8177, 80eqtrd 2833 . . . . . . . . . . . . . . 15 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) = (dom 𝑅 ∪ ran 𝑅))
8273, 81uneq12d 4067 . . . . . . . . . . . . . 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 4055 . . . . . . . . . . . . . 14 ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
8482, 83syl6eq 2849 . . . . . . . . . . . . 13 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) = (dom 𝑅 ∪ ran 𝑅))
8584uneq1d 4065 . . . . . . . . . . . 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 14239 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℕ0𝑅𝑉) → dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
8786expcom 414 . . . . . . . . . . . . . . . . . 18 (𝑅𝑉 → (𝑥 ∈ ℕ0 → dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
8887ralrimiv 3150 . . . . . . . . . . . . . . . . 17 (𝑅𝑉 → ∀𝑥 ∈ ℕ0 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
8988ad2antll 725 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ∀𝑥 ∈ ℕ0 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
90 olc 863 . . . . . . . . . . . . . . . . . . . . 21 (𝑍 ⊆ ℕ0 → ({1} ⊆ ℕ0𝑍 ⊆ ℕ0))
9190ad2antrl 724 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({1} ⊆ ℕ0𝑍 ⊆ ℕ0))
92 inss 4141 . . . . . . . . . . . . . . . . . . . 20 (({1} ⊆ ℕ0𝑍 ⊆ ℕ0) → ({1} ∩ 𝑍) ⊆ ℕ0)
9391, 92syl 17 . . . . . . . . . . . . . . . . . . 19 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({1} ∩ 𝑍) ⊆ ℕ0)
9493sseld 3894 . . . . . . . . . . . . . . . . . 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 3145 . . . . . . . . . . . . . . . 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 4874 . . . . . . . . . . . . . . 15 ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
9997, 98sylibr 235 . . . . . . . . . . . . . 14 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
100 relexprng 14243 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℕ0𝑅𝑉) → ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
101100expcom 414 . . . . . . . . . . . . . . . . . 18 (𝑅𝑉 → (𝑥 ∈ ℕ0 → ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
102101ralrimiv 3150 . . . . . . . . . . . . . . . . 17 (𝑅𝑉 → ∀𝑥 ∈ ℕ0 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
103102ad2antll 725 . . . . . . . . . . . . . . . 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 3145 . . . . . . . . . . . . . . . 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 4874 . . . . . . . . . . . . . . 15 ( 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
108106, 107sylibr 235 . . . . . . . . . . . . . 14 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
10999, 108unssd 4089 . . . . . . . . . . . . 13 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
110 ssequn2 4086 . . . . . . . . . . . . 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 219 . . . . . . . . . . . 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 2833 . . . . . . . . . . 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 413 . . . . . . . . . 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 483 . . . . . . . . . . . . . . . . . . 19 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 1 ∈ 𝑍)
115114snssd 4655 . . . . . . . . . . . . . . . . . 18 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → {1} ⊆ 𝑍)
116 df-ss 3880 . . . . . . . . . . . . . . . . . 18 ({1} ⊆ 𝑍 ↔ ({1} ∩ 𝑍) = {1})
117115, 116sylib 219 . . . . . . . . . . . . . . . . 17 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({1} ∩ 𝑍) = {1})
118117iuneq1d 4857 . . . . . . . . . . . . . . . 16 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) = 𝑥 ∈ {1}dom (𝑅𝑟𝑥))
119 1ex 10490 . . . . . . . . . . . . . . . . 17 1 ∈ V
120 oveq2 7031 . . . . . . . . . . . . . . . . . 18 (𝑥 = 1 → (𝑅𝑟𝑥) = (𝑅𝑟1))
121120dmeqd 5667 . . . . . . . . . . . . . . . . 17 (𝑥 = 1 → dom (𝑅𝑟𝑥) = dom (𝑅𝑟1))
122119, 121iunxsn 4918 . . . . . . . . . . . . . . . 16 𝑥 ∈ {1}dom (𝑅𝑟𝑥) = dom (𝑅𝑟1)
123118, 122syl6eq 2849 . . . . . . . . . . . . . . 15 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) = dom (𝑅𝑟1))
124 relexp1g 14223 . . . . . . . . . . . . . . . . 17 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
125124ad2antll 725 . . . . . . . . . . . . . . . 16 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (𝑅𝑟1) = 𝑅)
126125dmeqd 5667 . . . . . . . . . . . . . . 15 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → dom (𝑅𝑟1) = dom 𝑅)
127123, 126eqtrd 2833 . . . . . . . . . . . . . 14 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) = dom 𝑅)
128117iuneq1d 4857 . . . . . . . . . . . . . . . 16 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) = 𝑥 ∈ {1}ran (𝑅𝑟𝑥))
129120rneqd 5697 . . . . . . . . . . . . . . . . 17 (𝑥 = 1 → ran (𝑅𝑟𝑥) = ran (𝑅𝑟1))
130119, 129iunxsn 4918 . . . . . . . . . . . . . . . 16 𝑥 ∈ {1}ran (𝑅𝑟𝑥) = ran (𝑅𝑟1)
131128, 130syl6eq 2849 . . . . . . . . . . . . . . 15 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) = ran (𝑅𝑟1))
132125rneqd 5697 . . . . . . . . . . . . . . 15 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ran (𝑅𝑟1) = ran 𝑅)
133131, 132eqtrd 2833 . . . . . . . . . . . . . 14 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) = ran 𝑅)
134127, 133uneq12d 4067 . . . . . . . . . . . . 13 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) = (dom 𝑅 ∪ ran 𝑅))
135134uneq2d 4066 . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . 16 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ∀𝑥 ∈ ℕ0 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
137 olc 863 . . . . . . . . . . . . . . . . . . . . 21 (𝑍 ⊆ ℕ0 → ({0} ⊆ ℕ0𝑍 ⊆ ℕ0))
138137ad2antrl 724 . . . . . . . . . . . . . . . . . . . 20 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({0} ⊆ ℕ0𝑍 ⊆ ℕ0))
139 inss 4141 . . . . . . . . . . . . . . . . . . . 20 (({0} ⊆ ℕ0𝑍 ⊆ ℕ0) → ({0} ∩ 𝑍) ⊆ ℕ0)
140138, 139syl 17 . . . . . . . . . . . . . . . . . . 19 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({0} ∩ 𝑍) ⊆ ℕ0)
141140sseld 3894 . . . . . . . . . . . . . . . . . 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 3145 . . . . . . . . . . . . . . . 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 4874 . . . . . . . . . . . . . . 15 ( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
146144, 145sylibr 235 . . . . . . . . . . . . . 14 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
147102ad2antll 725 . . . . . . . . . . . . . . . 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 3145 . . . . . . . . . . . . . . . 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 4874 . . . . . . . . . . . . . . 15 ( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
152150, 151sylibr 235 . . . . . . . . . . . . . 14 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
153146, 152unssd 4089 . . . . . . . . . . . . 13 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
154 ssequn1 4083 . . . . . . . . . . . . 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 219 . . . . . . . . . . . 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 2833 . . . . . . . . . . 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 413 . . . . . . . . . 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 852 . . . . . . . . 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 1109 . . . . . . 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 1117 . . . . . 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 4065 . . . . 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 481 . . . . . . . . . 10 ((𝑅𝑉𝑍 ⊆ ℕ0) → ∀𝑥 ∈ ℕ0 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
164 ssel 3889 . . . . . . . . . . . . 13 (𝑍 ⊆ ℕ0 → (𝑥𝑍𝑥 ∈ ℕ0))
165164adantl 482 . . . . . . . . . . . 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 3145 . . . . . . . . . 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 4874 . . . . . . . . 9 ( 𝑥𝑍 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥𝑍 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
170168, 169sylibr 235 . . . . . . . 8 ((𝑅𝑉𝑍 ⊆ ℕ0) → 𝑥𝑍 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
171102adantr 481 . . . . . . . . . 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 3145 . . . . . . . . . 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 4874 . . . . . . . . 9 ( 𝑥𝑍 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥𝑍 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
176174, 175sylibr 235 . . . . . . . 8 ((𝑅𝑉𝑍 ⊆ ℕ0) → 𝑥𝑍 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
177170, 176unssd 4089 . . . . . . 7 ((𝑅𝑉𝑍 ⊆ ℕ0) → ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
1781773adant3 1125 . . . . . 6 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
179 ssequn2 4086 . . . . . 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 219 . . . . 5 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ((dom 𝑅 ∪ ran 𝑅) ∪ ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
181162, 180eqtrd 2833 . . . 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, 181syl5eq 2845 . . 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 11757 . . . . . . 7 0 ∈ V
184183ssex 5123 . . . . . 6 (𝑍 ⊆ ℕ0𝑍 ∈ V)
185 incom 4105 . . . . . . . . . 10 (𝑍 ∩ {0}) = ({0} ∩ 𝑍)
186 inex1g 5121 . . . . . . . . . 10 (𝑍 ∈ V → (𝑍 ∩ {0}) ∈ V)
187185, 186syl5eqelr 2890 . . . . . . . . 9 (𝑍 ∈ V → ({0} ∩ 𝑍) ∈ V)
188 incom 4105 . . . . . . . . . 10 (𝑍 ∩ {1}) = ({1} ∩ 𝑍)
189 inex1g 5121 . . . . . . . . . 10 (𝑍 ∈ V → (𝑍 ∩ {1}) ∈ V)
190188, 189syl5eqelr 2890 . . . . . . . . 9 (𝑍 ∈ V → ({1} ∩ 𝑍) ∈ V)
191 unexg 7336 . . . . . . . . 9 ((({0} ∩ 𝑍) ∈ V ∧ ({1} ∩ 𝑍) ∈ V) → (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∈ V)
192187, 190, 191syl2anc 584 . . . . . . . 8 (𝑍 ∈ V → (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∈ V)
193 unexg 7336 . . . . . . . 8 (((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∈ V ∧ 𝑍 ∈ V) → ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) ∈ V)
194192, 193mpancom 684 . . . . . . 7 (𝑍 ∈ V → ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) ∈ V)
195 ovex 7055 . . . . . . . 8 (𝑅𝑟𝑥) ∈ V
196195rgenw 3119 . . . . . . 7 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∈ V
197 iunexg 7527 . . . . . . 7 ((((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) ∈ V ∧ ∀𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∈ V) → 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∈ V)
198194, 196, 197sylancl 586 . . . . . 6 (𝑍 ∈ V → 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∈ V)
199184, 198syl 17 . . . . 5 (𝑍 ⊆ ℕ0 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∈ V)
2001993ad2ant2 1127 . . . 4 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∈ V)
201 simp1 1129 . . . 4 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → 𝑅𝑉)
202 relexp0eq 39552 . . . 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)))
203200, 201, 202syl2anc 584 . . 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)))
204182, 203mpbid 233 . 2 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ( 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥)↑𝑟0) = (𝑅𝑟0))
20512, 204syl5eq 2845 1 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ( 𝑥𝑍 (𝑅𝑟𝑥)↑𝑟0) = (𝑅𝑟0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 842  w3a 1080   = wceq 1525  wcel 2083  wne 2986  wral 3107  Vcvv 3440  cun 3863  cin 3864  wss 3865  c0 4217  {csn 4478  {cpr 4480   ciun 4831   I cid 5354  dom cdm 5450  ran crn 5451  cres 5452  (class class class)co 7023  0cc0 10390  1c1 10391  0cn0 11751  𝑟crelexp 14217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-er 8146  df-en 8365  df-dom 8366  df-sdom 8367  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-nn 11493  df-n0 11752  df-z 11836  df-uz 12098  df-seq 13224  df-relexp 14218
This theorem is referenced by:  corclrcl  39558  corcltrcl  39590
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