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Theorem relexpfld 14015
Description: The field of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
Assertion
Ref Expression
relexpfld ((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) ⊆ 𝑅)

Proof of Theorem relexpfld
StepHypRef Expression
1 simpl 470 . . . . . . . 8 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → 𝑁 = 1)
21oveq2d 6893 . . . . . . 7 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟𝑁) = (𝑅𝑟1))
3 relexp1g 13992 . . . . . . . 8 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
43ad2antll 711 . . . . . . 7 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟1) = 𝑅)
52, 4eqtrd 2847 . . . . . 6 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟𝑁) = 𝑅)
65unieqd 4647 . . . . 5 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟𝑁) = 𝑅)
76unieqd 4647 . . . 4 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟𝑁) = 𝑅)
8 eqimss 3861 . . . 4 ( (𝑅𝑟𝑁) = 𝑅 (𝑅𝑟𝑁) ⊆ 𝑅)
97, 8syl 17 . . 3 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟𝑁) ⊆ 𝑅)
109ex 399 . 2 (𝑁 = 1 → ((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) ⊆ 𝑅))
11 simp2 1160 . . . . . . 7 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → 𝑁 ∈ ℕ0)
12 simp3 1161 . . . . . . 7 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → 𝑅𝑉)
13 simp1 1159 . . . . . . . 8 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → ¬ 𝑁 = 1)
1413pm2.21d 119 . . . . . . 7 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → (𝑁 = 1 → Rel 𝑅))
1511, 12, 143jca 1151 . . . . . 6 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → (𝑁 ∈ ℕ0𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)))
16 relexprelg 14004 . . . . . 6 ((𝑁 ∈ ℕ0𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁))
17 relfld 5882 . . . . . 6 (Rel (𝑅𝑟𝑁) → (𝑅𝑟𝑁) = (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)))
1815, 16, 173syl 18 . . . . 5 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) = (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)))
19 elnn0 11564 . . . . . . 7 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
20 relexpnndm 14007 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ dom 𝑅)
21 relexpnnrn 14011 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑅𝑉) → ran (𝑅𝑟𝑁) ⊆ ran 𝑅)
22 unss12 3991 . . . . . . . . . 10 ((dom (𝑅𝑟𝑁) ⊆ dom 𝑅 ∧ ran (𝑅𝑟𝑁) ⊆ ran 𝑅) → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
2320, 21, 22syl2anc 575 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑅𝑉) → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
2423ex 399 . . . . . . . 8 (𝑁 ∈ ℕ → (𝑅𝑉 → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
25 simpl 470 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑅𝑉) → 𝑁 = 0)
2625oveq2d 6893 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
27 relexp0g 13988 . . . . . . . . . . . . . . 15 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
2827adantl 469 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
2926, 28eqtrd 2847 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
3029dmeqd 5534 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) = dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
31 dmresi 5676 . . . . . . . . . . . 12 dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
3230, 31syl6eq 2863 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅))
33 eqimss 3861 . . . . . . . . . . 11 (dom (𝑅𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅) → dom (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
3432, 33syl 17 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
3529rneqd 5561 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑅𝑉) → ran (𝑅𝑟𝑁) = ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
36 rnresi 5696 . . . . . . . . . . . 12 ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
3735, 36syl6eq 2863 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑅𝑉) → ran (𝑅𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅))
38 eqimss 3861 . . . . . . . . . . 11 (ran (𝑅𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅) → ran (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
3937, 38syl 17 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑅𝑉) → ran (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
4034, 39unssd 3995 . . . . . . . . 9 ((𝑁 = 0 ∧ 𝑅𝑉) → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
4140ex 399 . . . . . . . 8 (𝑁 = 0 → (𝑅𝑉 → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
4224, 41jaoi 875 . . . . . . 7 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑅𝑉 → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
4319, 42sylbi 208 . . . . . 6 (𝑁 ∈ ℕ0 → (𝑅𝑉 → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
4411, 12, 43sylc 65 . . . . 5 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
4518, 44eqsstrd 3843 . . . 4 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
46 dmrnssfld 5592 . . . 4 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
4745, 46syl6ss 3817 . . 3 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) ⊆ 𝑅)
48473expib 1145 . 2 𝑁 = 1 → ((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) ⊆ 𝑅))
4910, 48pm2.61i 176 1 ((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) ⊆ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wo 865  w3a 1100   = wceq 1637  wcel 2157  cun 3774  wss 3776   cuni 4637   I cid 5225  dom cdm 5318  ran crn 5319  cres 5320  Rel wrel 5323  (class class class)co 6877  0cc0 10224  1c1 10225  cn 11308  0cn0 11562  𝑟crelexp 13986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-om 7299  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-pnf 10364  df-mnf 10365  df-xr 10366  df-ltxr 10367  df-le 10368  df-sub 10556  df-neg 10557  df-nn 11309  df-n0 11563  df-z 11647  df-uz 11908  df-seq 13028  df-relexp 13987
This theorem is referenced by:  relexpfldd  14016
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