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Mirrors > Home > MPE Home > Th. List > relexpnndm | 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 |
---|---|
relexpnndm | ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 6557 | . . . . . 6 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1)) | |
2 | 1 | dmeqd 5248 | . . . . 5 ⊢ (𝑛 = 1 → dom (𝑅↑𝑟𝑛) = dom (𝑅↑𝑟1)) |
3 | 2 | sseq1d 3595 | . . . 4 ⊢ (𝑛 = 1 → (dom (𝑅↑𝑟𝑛) ⊆ dom 𝑅 ↔ dom (𝑅↑𝑟1) ⊆ dom 𝑅)) |
4 | 3 | imbi2d 329 | . . 3 ⊢ (𝑛 = 1 → ((𝑅 ∈ 𝑉 → dom (𝑅↑𝑟𝑛) ⊆ dom 𝑅) ↔ (𝑅 ∈ 𝑉 → dom (𝑅↑𝑟1) ⊆ dom 𝑅))) |
5 | oveq2 6557 | . . . . . 6 ⊢ (𝑛 = 𝑚 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑚)) | |
6 | 5 | dmeqd 5248 | . . . . 5 ⊢ (𝑛 = 𝑚 → dom (𝑅↑𝑟𝑛) = dom (𝑅↑𝑟𝑚)) |
7 | 6 | sseq1d 3595 | . . . 4 ⊢ (𝑛 = 𝑚 → (dom (𝑅↑𝑟𝑛) ⊆ dom 𝑅 ↔ dom (𝑅↑𝑟𝑚) ⊆ dom 𝑅)) |
8 | 7 | imbi2d 329 | . . 3 ⊢ (𝑛 = 𝑚 → ((𝑅 ∈ 𝑉 → dom (𝑅↑𝑟𝑛) ⊆ dom 𝑅) ↔ (𝑅 ∈ 𝑉 → dom (𝑅↑𝑟𝑚) ⊆ dom 𝑅))) |
9 | oveq2 6557 | . . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1))) | |
10 | 9 | dmeqd 5248 | . . . . 5 ⊢ (𝑛 = (𝑚 + 1) → dom (𝑅↑𝑟𝑛) = dom (𝑅↑𝑟(𝑚 + 1))) |
11 | 10 | sseq1d 3595 | . . . 4 ⊢ (𝑛 = (𝑚 + 1) → (dom (𝑅↑𝑟𝑛) ⊆ dom 𝑅 ↔ dom (𝑅↑𝑟(𝑚 + 1)) ⊆ dom 𝑅)) |
12 | 11 | imbi2d 329 | . . 3 ⊢ (𝑛 = (𝑚 + 1) → ((𝑅 ∈ 𝑉 → dom (𝑅↑𝑟𝑛) ⊆ dom 𝑅) ↔ (𝑅 ∈ 𝑉 → dom (𝑅↑𝑟(𝑚 + 1)) ⊆ dom 𝑅))) |
13 | oveq2 6557 | . . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑁)) | |
14 | 13 | dmeqd 5248 | . . . . 5 ⊢ (𝑛 = 𝑁 → dom (𝑅↑𝑟𝑛) = dom (𝑅↑𝑟𝑁)) |
15 | 14 | sseq1d 3595 | . . . 4 ⊢ (𝑛 = 𝑁 → (dom (𝑅↑𝑟𝑛) ⊆ dom 𝑅 ↔ dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅)) |
16 | 15 | imbi2d 329 | . . 3 ⊢ (𝑛 = 𝑁 → ((𝑅 ∈ 𝑉 → dom (𝑅↑𝑟𝑛) ⊆ dom 𝑅) ↔ (𝑅 ∈ 𝑉 → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅))) |
17 | relexp1g 13614 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅) | |
18 | 17 | dmeqd 5248 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → dom (𝑅↑𝑟1) = dom 𝑅) |
19 | eqimss 3620 | . . . 4 ⊢ (dom (𝑅↑𝑟1) = dom 𝑅 → dom (𝑅↑𝑟1) ⊆ dom 𝑅) | |
20 | 18, 19 | syl 17 | . . 3 ⊢ (𝑅 ∈ 𝑉 → dom (𝑅↑𝑟1) ⊆ dom 𝑅) |
21 | relexpsucnnr 13613 | . . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅)) | |
22 | 21 | ancoms 468 | . . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅)) |
23 | 22 | dmeqd 5248 | . . . . . . 7 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟(𝑚 + 1)) = dom ((𝑅↑𝑟𝑚) ∘ 𝑅)) |
24 | dmcoss 5306 | . . . . . . 7 ⊢ dom ((𝑅↑𝑟𝑚) ∘ 𝑅) ⊆ dom 𝑅 | |
25 | 23, 24 | syl6eqss 3618 | . . . . . 6 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟(𝑚 + 1)) ⊆ dom 𝑅) |
26 | 25 | a1d 25 | . . . . 5 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (dom (𝑅↑𝑟𝑚) ⊆ dom 𝑅 → dom (𝑅↑𝑟(𝑚 + 1)) ⊆ dom 𝑅)) |
27 | 26 | ex 449 | . . . 4 ⊢ (𝑚 ∈ ℕ → (𝑅 ∈ 𝑉 → (dom (𝑅↑𝑟𝑚) ⊆ dom 𝑅 → dom (𝑅↑𝑟(𝑚 + 1)) ⊆ dom 𝑅))) |
28 | 27 | a2d 29 | . . 3 ⊢ (𝑚 ∈ ℕ → ((𝑅 ∈ 𝑉 → dom (𝑅↑𝑟𝑚) ⊆ dom 𝑅) → (𝑅 ∈ 𝑉 → dom (𝑅↑𝑟(𝑚 + 1)) ⊆ dom 𝑅))) |
29 | 4, 8, 12, 16, 20, 28 | nnind 10915 | . 2 ⊢ (𝑁 ∈ ℕ → (𝑅 ∈ 𝑉 → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅)) |
30 | 29 | imp 444 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 dom cdm 5038 ∘ ccom 5042 (class class class)co 6549 1c1 9816 + caddc 9818 ℕcn 10897 ↑𝑟crelexp 13608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-seq 12664 df-relexp 13609 |
This theorem is referenced by: relexpdmg 13630 relexpnnrn 13633 relexpfld 13637 relexpaddg 13641 relexpaddss 37029 |
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