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Theorem relexp01min 38322
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 4230 . . 3 (𝐽 ∈ {0, 1} → (𝐽 = 0 ∨ 𝐽 = 1))
2 elpri 4230 . . 3 (𝐾 ∈ {0, 1} → (𝐾 = 0 ∨ 𝐾 = 1))
3 dmresi 5492 . . . . . . . . . . 11 dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
4 rnresi 5514 . . . . . . . . . . 11 ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
53, 4uneq12i 3798 . . . . . . . . . 10 (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅))
6 unidm 3789 . . . . . . . . . 10 ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
75, 6eqtri 2673 . . . . . . . . 9 (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (dom 𝑅 ∪ ran 𝑅)
87reseq2i 5425 . . . . . . . 8 ( I ↾ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
9 simp1 1081 . . . . . . . . . . . 12 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 0)
109oveq2d 6706 . . . . . . . . . . 11 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟0))
11 simp3l 1109 . . . . . . . . . . . 12 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅𝑉)
12 relexp0g 13806 . . . . . . . . . . . 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 2685 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
15 simp2 1082 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 0)
1614, 15oveq12d 6708 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟0))
17 dmexg 7139 . . . . . . . . . . . 12 (𝑅𝑉 → dom 𝑅 ∈ V)
18 rnexg 7140 . . . . . . . . . . . 12 (𝑅𝑉 → ran 𝑅 ∈ V)
19 unexg 7001 . . . . . . . . . . . 12 ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
2017, 18, 19syl2anc 694 . . . . . . . . . . 11 (𝑅𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
2120resiexd 6521 . . . . . . . . . 10 (𝑅𝑉 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
22 relexp0g 13806 . . . . . . . . . 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)))))
2311, 21, 223syl 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)))))
2416, 23eqtrd 2685 . . . . . . . 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)))))
25 simp3r 1110 . . . . . . . . . . 11 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
26 0re 10078 . . . . . . . . . . . . . 14 0 ∈ ℝ
2726ltnri 10184 . . . . . . . . . . . . 13 ¬ 0 < 0
289, 15breq12d 4698 . . . . . . . . . . . . 13 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 0 < 0))
2927, 28mtbiri 316 . . . . . . . . . . . 12 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 𝐽 < 𝐾)
3029iffalsed 4130 . . . . . . . . . . 11 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
3125, 30, 153eqtrd 2689 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
3231oveq2d 6706 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟0))
3332, 13eqtrd 2685 . . . . . . . 8 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
348, 24, 333eqtr4a 2711 . . . . . . 7 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
35343exp 1283 . . . . . 6 (𝐽 = 0 → (𝐾 = 0 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
36 simp1 1081 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 1)
3736oveq2d 6706 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟1))
38 simp3l 1109 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅𝑉)
39 relexp1g 13810 . . . . . . . . . . 11 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
4038, 39syl 17 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟1) = 𝑅)
4137, 40eqtrd 2685 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = 𝑅)
42 simp2 1082 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 0)
4341, 42oveq12d 6708 . . . . . . . 8 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟0))
44 simp3r 1110 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
45 0lt1 10588 . . . . . . . . . . . . 13 0 < 1
46 1re 10077 . . . . . . . . . . . . . 14 1 ∈ ℝ
4726, 46ltnsymi 10194 . . . . . . . . . . . . 13 (0 < 1 → ¬ 1 < 0)
4845, 47mp1i 13 . . . . . . . . . . . 12 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 1 < 0)
4936, 42breq12d 4698 . . . . . . . . . . . 12 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 1 < 0))
5048, 49mtbird 314 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 𝐽 < 𝐾)
5150iffalsed 4130 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
5244, 51, 423eqtrd 2689 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
5352oveq2d 6706 . . . . . . . 8 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟0))
5443, 53eqtr4d 2688 . . . . . . 7 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
55543exp 1283 . . . . . 6 (𝐽 = 1 → (𝐾 = 0 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
5635, 55jaoi 393 . . . . 5 ((𝐽 = 0 ∨ 𝐽 = 1) → (𝐾 = 0 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
57 ovex 6718 . . . . . . . . 9 (𝑅𝑟0) ∈ V
58 relexp1g 13810 . . . . . . . . 9 ((𝑅𝑟0) ∈ V → ((𝑅𝑟0)↑𝑟1) = (𝑅𝑟0))
5957, 58mp1i 13 . . . . . . . 8 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟0)↑𝑟1) = (𝑅𝑟0))
60 simp1 1081 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 0)
6160oveq2d 6706 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟0))
62 simp2 1082 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 1)
6361, 62oveq12d 6708 . . . . . . . 8 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = ((𝑅𝑟0)↑𝑟1))
64 simp3r 1110 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
6560, 62breq12d 4698 . . . . . . . . . . . 12 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 0 < 1))
6645, 65mpbiri 248 . . . . . . . . . . 11 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 < 𝐾)
6766iftrued 4127 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽)
6864, 67, 603eqtrd 2689 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
6968oveq2d 6706 . . . . . . . 8 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟0))
7059, 63, 693eqtr4d 2695 . . . . . . 7 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
71703exp 1283 . . . . . 6 (𝐽 = 0 → (𝐾 = 1 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
72 simp1 1081 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 1)
7372oveq2d 6706 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟1))
74 simp3l 1109 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅𝑉)
7574, 39syl 17 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟1) = 𝑅)
7673, 75eqtrd 2685 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = 𝑅)
77 simp2 1082 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 1)
7876, 77oveq12d 6708 . . . . . . . 8 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟1))
79 simp3r 1110 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
8046ltnri 10184 . . . . . . . . . . . 12 ¬ 1 < 1
8172, 77breq12d 4698 . . . . . . . . . . . 12 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 1 < 1))
8280, 81mtbiri 316 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 𝐽 < 𝐾)
8382iffalsed 4130 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
8479, 83, 773eqtrd 2689 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 1)
8584oveq2d 6706 . . . . . . . 8 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟1))
8678, 85eqtr4d 2688 . . . . . . 7 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
87863exp 1283 . . . . . 6 (𝐽 = 1 → (𝐾 = 1 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
8871, 87jaoi 393 . . . . 5 ((𝐽 = 0 ∨ 𝐽 = 1) → (𝐾 = 1 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
8956, 88jaod 394 . . . 4 ((𝐽 = 0 ∨ 𝐽 = 1) → ((𝐾 = 0 ∨ 𝐾 = 1) → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
9089imp 444 . . 3 (((𝐽 = 0 ∨ 𝐽 = 1) ∧ (𝐾 = 0 ∨ 𝐾 = 1)) → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼)))
911, 2, 90syl2an 493 . 2 ((𝐽 ∈ {0, 1} ∧ 𝐾 ∈ {0, 1}) → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼)))
9291impcom 445 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  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  Vcvv 3231  cun 3605  ifcif 4119  {cpr 4212   class class class wbr 4685   I cid 5052  dom cdm 5143  ran crn 5144  cres 5145  (class class class)co 6690  0cc0 9974  1c1 9975   < clt 10112  𝑟crelexp 13804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-seq 12842  df-relexp 13805
This theorem is referenced by:  relexp1idm  38323  relexp0idm  38324
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