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Theorem relexp01min 43726
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 4649 . . 3 (𝐽 ∈ {0, 1} → (𝐽 = 0 ∨ 𝐽 = 1))
2 elpri 4649 . . 3 (𝐾 ∈ {0, 1} → (𝐾 = 0 ∨ 𝐾 = 1))
3 dmresi 6070 . . . . . . . . . . 11 dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
4 rnresi 6093 . . . . . . . . . . 11 ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
53, 4uneq12i 4166 . . . . . . . . . 10 (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅))
6 unidm 4157 . . . . . . . . . 10 ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
75, 6eqtri 2765 . . . . . . . . 9 (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (dom 𝑅 ∪ ran 𝑅)
87reseq2i 5994 . . . . . . . 8 ( I ↾ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
9 simp1 1137 . . . . . . . . . . . 12 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 0)
109oveq2d 7447 . . . . . . . . . . 11 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟0))
11 simp3l 1202 . . . . . . . . . . . 12 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅𝑉)
12 relexp0g 15061 . . . . . . . . . . . 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 2777 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
15 simp2 1138 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 0)
1614, 15oveq12d 7449 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟0))
17 dmexg 7923 . . . . . . . . . . . 12 (𝑅𝑉 → dom 𝑅 ∈ V)
18 rnexg 7924 . . . . . . . . . . . 12 (𝑅𝑉 → ran 𝑅 ∈ V)
1917, 18unexd 7774 . . . . . . . . . . 11 (𝑅𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
2019resiexd 7236 . . . . . . . . . 10 (𝑅𝑉 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
21 relexp0g 15061 . . . . . . . . . 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 2777 . . . . . . . 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 1203 . . . . . . . . . . 11 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
25 0re 11263 . . . . . . . . . . . . . 14 0 ∈ ℝ
2625ltnri 11370 . . . . . . . . . . . . 13 ¬ 0 < 0
279, 15breq12d 5156 . . . . . . . . . . . . 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 4536 . . . . . . . . . . 11 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
3024, 29, 153eqtrd 2781 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
3130oveq2d 7447 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟0))
3231, 13eqtrd 2777 . . . . . . . 8 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
338, 23, 323eqtr4a 2803 . . . . . . 7 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
34333exp 1120 . . . . . 6 (𝐽 = 0 → (𝐾 = 0 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
35 simp1 1137 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 1)
3635oveq2d 7447 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟1))
37 simp3l 1202 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅𝑉)
3837relexp1d 15068 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟1) = 𝑅)
3936, 38eqtrd 2777 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = 𝑅)
40 simp2 1138 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 0)
4139, 40oveq12d 7449 . . . . . . . 8 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟0))
42 simp3r 1203 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
43 0lt1 11785 . . . . . . . . . . . . 13 0 < 1
44 1re 11261 . . . . . . . . . . . . . 14 1 ∈ ℝ
4525, 44ltnsymi 11380 . . . . . . . . . . . . 13 (0 < 1 → ¬ 1 < 0)
4643, 45mp1i 13 . . . . . . . . . . . 12 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 1 < 0)
4735, 40breq12d 5156 . . . . . . . . . . . 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 4536 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
5042, 49, 403eqtrd 2781 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
5150oveq2d 7447 . . . . . . . 8 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟0))
5241, 51eqtr4d 2780 . . . . . . 7 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
53523exp 1120 . . . . . 6 (𝐽 = 1 → (𝐾 = 0 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
5434, 53jaoi 858 . . . . 5 ((𝐽 = 0 ∨ 𝐽 = 1) → (𝐾 = 0 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
55 ovex 7464 . . . . . . . . 9 (𝑅𝑟0) ∈ V
56 relexp1g 15065 . . . . . . . . 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 1137 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 0)
5958oveq2d 7447 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟0))
60 simp2 1138 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 1)
6159, 60oveq12d 7449 . . . . . . . 8 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = ((𝑅𝑟0)↑𝑟1))
62 simp3r 1203 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
6358, 60breq12d 5156 . . . . . . . . . . . 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 4533 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽)
6662, 65, 583eqtrd 2781 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
6766oveq2d 7447 . . . . . . . 8 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟0))
6857, 61, 673eqtr4d 2787 . . . . . . 7 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
69683exp 1120 . . . . . 6 (𝐽 = 0 → (𝐾 = 1 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
70 simp1 1137 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 1)
7170oveq2d 7447 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟1))
72 simp3l 1202 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅𝑉)
7372relexp1d 15068 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟1) = 𝑅)
7471, 73eqtrd 2777 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = 𝑅)
75 simp2 1138 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 1)
7674, 75oveq12d 7449 . . . . . . . 8 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟1))
77 simp3r 1203 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
7844ltnri 11370 . . . . . . . . . . . 12 ¬ 1 < 1
7970, 75breq12d 5156 . . . . . . . . . . . 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 4536 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
8277, 81, 753eqtrd 2781 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 1)
8382oveq2d 7447 . . . . . . . 8 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟1))
8476, 83eqtr4d 2780 . . . . . . 7 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
85843exp 1120 . . . . . 6 (𝐽 = 1 → (𝐾 = 1 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
8669, 85jaoi 858 . . . . 5 ((𝐽 = 0 ∨ 𝐽 = 1) → (𝐾 = 1 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
8754, 86jaod 860 . . . 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 596 . 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 848  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  cun 3949  ifcif 4525  {cpr 4628   class class class wbr 5143   I cid 5577  dom cdm 5685  ran crn 5686  cres 5687  (class class class)co 7431  0cc0 11155  1c1 11156   < clt 11295  𝑟crelexp 15058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-seq 14043  df-relexp 15059
This theorem is referenced by:  relexp1idm  43727  relexp0idm  43728
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