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Theorem relexp01min 43056
Description: With exponents limited to 0 and 1, the composition of powers of a relation is the relation raised to the minimum of exponents. (Contributed by RP, 12-Jun-2020.)
Assertion
Ref Expression
relexp01min (((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) ∧ (𝐽 ∈ {0, 1} ∧ 𝐾 ∈ {0, 1})) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))

Proof of Theorem relexp01min
StepHypRef Expression
1 elpri 4646 . . 3 (𝐽 ∈ {0, 1} → (𝐽 = 0 ∨ 𝐽 = 1))
2 elpri 4646 . . 3 (𝐾 ∈ {0, 1} → (𝐾 = 0 ∨ 𝐾 = 1))
3 dmresi 6049 . . . . . . . . . . 11 dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
4 rnresi 6072 . . . . . . . . . . 11 ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
53, 4uneq12i 4157 . . . . . . . . . 10 (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅))
6 unidm 4148 . . . . . . . . . 10 ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
75, 6eqtri 2755 . . . . . . . . 9 (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (dom 𝑅 ∪ ran 𝑅)
87reseq2i 5976 . . . . . . . 8 ( I ↾ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
9 simp1 1134 . . . . . . . . . . . 12 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 0)
109oveq2d 7430 . . . . . . . . . . 11 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟0))
11 simp3l 1199 . . . . . . . . . . . 12 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅𝑉)
12 relexp0g 14987 . . . . . . . . . . . 12 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
1311, 12syl 17 . . . . . . . . . . 11 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
1410, 13eqtrd 2767 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
15 simp2 1135 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 0)
1614, 15oveq12d 7432 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟0))
17 dmexg 7901 . . . . . . . . . . . 12 (𝑅𝑉 → dom 𝑅 ∈ V)
18 rnexg 7902 . . . . . . . . . . . 12 (𝑅𝑉 → ran 𝑅 ∈ V)
1917, 18unexd 7748 . . . . . . . . . . 11 (𝑅𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
2019resiexd 7222 . . . . . . . . . 10 (𝑅𝑉 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
21 relexp0g 14987 . . . . . . . . . 10 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V → (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟0) = ( I ↾ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))))
2211, 20, 213syl 18 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟0) = ( I ↾ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))))
2316, 22eqtrd 2767 . . . . . . . 8 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = ( I ↾ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))))
24 simp3r 1200 . . . . . . . . . . 11 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
25 0re 11232 . . . . . . . . . . . . . 14 0 ∈ ℝ
2625ltnri 11339 . . . . . . . . . . . . 13 ¬ 0 < 0
279, 15breq12d 5155 . . . . . . . . . . . . 13 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 0 < 0))
2826, 27mtbiri 327 . . . . . . . . . . . 12 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 𝐽 < 𝐾)
2928iffalsed 4535 . . . . . . . . . . 11 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
3024, 29, 153eqtrd 2771 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
3130oveq2d 7430 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟0))
3231, 13eqtrd 2767 . . . . . . . 8 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
338, 23, 323eqtr4a 2793 . . . . . . 7 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
34333exp 1117 . . . . . 6 (𝐽 = 0 → (𝐾 = 0 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
35 simp1 1134 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 1)
3635oveq2d 7430 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟1))
37 simp3l 1199 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅𝑉)
3837relexp1d 14994 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟1) = 𝑅)
3936, 38eqtrd 2767 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = 𝑅)
40 simp2 1135 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 0)
4139, 40oveq12d 7432 . . . . . . . 8 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟0))
42 simp3r 1200 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
43 0lt1 11752 . . . . . . . . . . . . 13 0 < 1
44 1re 11230 . . . . . . . . . . . . . 14 1 ∈ ℝ
4525, 44ltnsymi 11349 . . . . . . . . . . . . 13 (0 < 1 → ¬ 1 < 0)
4643, 45mp1i 13 . . . . . . . . . . . 12 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 1 < 0)
4735, 40breq12d 5155 . . . . . . . . . . . 12 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 1 < 0))
4846, 47mtbird 325 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 𝐽 < 𝐾)
4948iffalsed 4535 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
5042, 49, 403eqtrd 2771 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
5150oveq2d 7430 . . . . . . . 8 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟0))
5241, 51eqtr4d 2770 . . . . . . 7 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
53523exp 1117 . . . . . 6 (𝐽 = 1 → (𝐾 = 0 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
5434, 53jaoi 856 . . . . 5 ((𝐽 = 0 ∨ 𝐽 = 1) → (𝐾 = 0 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
55 ovex 7447 . . . . . . . . 9 (𝑅𝑟0) ∈ V
56 relexp1g 14991 . . . . . . . . 9 ((𝑅𝑟0) ∈ V → ((𝑅𝑟0)↑𝑟1) = (𝑅𝑟0))
5755, 56mp1i 13 . . . . . . . 8 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟0)↑𝑟1) = (𝑅𝑟0))
58 simp1 1134 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 0)
5958oveq2d 7430 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟0))
60 simp2 1135 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 1)
6159, 60oveq12d 7432 . . . . . . . 8 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = ((𝑅𝑟0)↑𝑟1))
62 simp3r 1200 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
6358, 60breq12d 5155 . . . . . . . . . . . 12 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 0 < 1))
6443, 63mpbiri 258 . . . . . . . . . . 11 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 < 𝐾)
6564iftrued 4532 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽)
6662, 65, 583eqtrd 2771 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
6766oveq2d 7430 . . . . . . . 8 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟0))
6857, 61, 673eqtr4d 2777 . . . . . . 7 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
69683exp 1117 . . . . . 6 (𝐽 = 0 → (𝐾 = 1 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
70 simp1 1134 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 1)
7170oveq2d 7430 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟1))
72 simp3l 1199 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅𝑉)
7372relexp1d 14994 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟1) = 𝑅)
7471, 73eqtrd 2767 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = 𝑅)
75 simp2 1135 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 1)
7674, 75oveq12d 7432 . . . . . . . 8 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟1))
77 simp3r 1200 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
7844ltnri 11339 . . . . . . . . . . . 12 ¬ 1 < 1
7970, 75breq12d 5155 . . . . . . . . . . . 12 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 1 < 1))
8078, 79mtbiri 327 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 𝐽 < 𝐾)
8180iffalsed 4535 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
8277, 81, 753eqtrd 2771 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 1)
8382oveq2d 7430 . . . . . . . 8 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟1))
8476, 83eqtr4d 2770 . . . . . . 7 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
85843exp 1117 . . . . . 6 (𝐽 = 1 → (𝐾 = 1 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
8669, 85jaoi 856 . . . . 5 ((𝐽 = 0 ∨ 𝐽 = 1) → (𝐾 = 1 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
8754, 86jaod 858 . . . 4 ((𝐽 = 0 ∨ 𝐽 = 1) → ((𝐾 = 0 ∨ 𝐾 = 1) → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
8887imp 406 . . 3 (((𝐽 = 0 ∨ 𝐽 = 1) ∧ (𝐾 = 0 ∨ 𝐾 = 1)) → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼)))
891, 2, 88syl2an 595 . 2 ((𝐽 ∈ {0, 1} ∧ 𝐾 ∈ {0, 1}) → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼)))
9089impcom 407 1 (((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) ∧ (𝐽 ∈ {0, 1} ∧ 𝐾 ∈ {0, 1})) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 846  w3a 1085   = wceq 1534  wcel 2099  Vcvv 3469  cun 3942  ifcif 4524  {cpr 4626   class class class wbr 5142   I cid 5569  dom cdm 5672  ran crn 5673  cres 5674  (class class class)co 7414  0cc0 11124  1c1 11125   < clt 11264  𝑟crelexp 14984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  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 5143  df-opab 5205  df-mpt 5226  df-tr 5260  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  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 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-er 8716  df-en 8954  df-dom 8955  df-sdom 8956  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-nn 12229  df-n0 12489  df-z 12575  df-uz 12839  df-seq 13985  df-relexp 14985
This theorem is referenced by:  relexp1idm  43057  relexp0idm  43058
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