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Mirrors > Home > MPE Home > Th. List > relexpdmg | Structured version Visualization version GIF version |
Description: The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.) |
Ref | Expression |
---|---|
relexpdmg | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 12478 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | relexpnndm 14994 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅) | |
3 | ssun1 4167 | . . . . . 6 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) | |
4 | 2, 3 | sstrdi 3989 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) |
5 | 4 | ex 412 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑅 ∈ 𝑉 → dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))) |
6 | simpl 482 | . . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0) | |
7 | 6 | oveq2d 7421 | . . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0)) |
8 | relexp0g 14975 | . . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) | |
9 | 8 | adantl 481 | . . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
10 | 7, 9 | eqtrd 2766 | . . . . . . . 8 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
11 | 10 | dmeqd 5899 | . . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) = dom ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
12 | dmresi 6045 | . . . . . . 7 ⊢ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅) | |
13 | 11, 12 | eqtrdi 2782 | . . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅)) |
14 | eqimss 4035 | . . . . . 6 ⊢ (dom (𝑅↑𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅) → dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) | |
15 | 13, 14 | syl 17 | . . . . 5 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) |
16 | 15 | ex 412 | . . . 4 ⊢ (𝑁 = 0 → (𝑅 ∈ 𝑉 → dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))) |
17 | 5, 16 | jaoi 854 | . . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑅 ∈ 𝑉 → dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))) |
18 | 1, 17 | sylbi 216 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑅 ∈ 𝑉 → dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))) |
19 | 18 | imp 406 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ∪ cun 3941 ⊆ wss 3943 I cid 5566 dom cdm 5669 ran crn 5670 ↾ cres 5671 (class class class)co 7405 0cc0 11112 ℕcn 12216 ℕ0cn0 12476 ↑𝑟crelexp 14972 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-seq 13973 df-relexp 14973 |
This theorem is referenced by: relexpdm 14996 iunrelexp0 43029 |
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