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Mirrors > Home > MPE Home > Th. List > relexprng | Structured version Visualization version GIF version |
Description: The range of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.) |
Ref | Expression |
---|---|
relexprng | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 12505 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | relexpnnrn 15025 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) ⊆ ran 𝑅) | |
3 | ssun2 4173 | . . . . . 6 ⊢ ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) | |
4 | 2, 3 | sstrdi 3992 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) |
5 | 4 | ex 412 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑅 ∈ 𝑉 → ran (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))) |
6 | simpl 482 | . . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0) | |
7 | 6 | oveq2d 7436 | . . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0)) |
8 | relexp0g 15002 | . . . . . . . . . 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 2768 | . . . . . . . 8 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
11 | 10 | rneqd 5940 | . . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) = ran ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
12 | rnresi 6078 | . . . . . . 7 ⊢ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅) | |
13 | 11, 12 | eqtrdi 2784 | . . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅)) |
14 | eqimss 4038 | . . . . . 6 ⊢ (ran (𝑅↑𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅) → ran (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) | |
15 | 13, 14 | syl 17 | . . . . 5 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) |
16 | 15 | ex 412 | . . . 4 ⊢ (𝑁 = 0 → (𝑅 ∈ 𝑉 → ran (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))) |
17 | 5, 16 | jaoi 856 | . . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑅 ∈ 𝑉 → ran (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))) |
18 | 1, 17 | sylbi 216 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑅 ∈ 𝑉 → ran (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))) |
19 | 18 | imp 406 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ∪ cun 3945 ⊆ wss 3947 I cid 5575 dom cdm 5678 ran crn 5679 ↾ cres 5680 (class class class)co 7420 0cc0 11139 ℕcn 12243 ℕ0cn0 12503 ↑𝑟crelexp 14999 |
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 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-seq 14000 df-relexp 15000 |
This theorem is referenced by: relexprn 15027 iunrelexp0 43132 |
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