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Theorem relexpfld 14408
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 486 . . . . . . . 8 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → 𝑁 = 1)
21oveq2d 7165 . . . . . . 7 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟𝑁) = (𝑅𝑟1))
3 relexp1g 14385 . . . . . . . 8 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
43ad2antll 728 . . . . . . 7 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟1) = 𝑅)
52, 4eqtrd 2859 . . . . . 6 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟𝑁) = 𝑅)
65unieqd 4838 . . . . 5 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟𝑁) = 𝑅)
76unieqd 4838 . . . 4 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟𝑁) = 𝑅)
8 eqimss 4009 . . . 4 ( (𝑅𝑟𝑁) = 𝑅 (𝑅𝑟𝑁) ⊆ 𝑅)
97, 8syl 17 . . 3 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟𝑁) ⊆ 𝑅)
109ex 416 . 2 (𝑁 = 1 → ((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) ⊆ 𝑅))
11 simp2 1134 . . . . . . 7 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → 𝑁 ∈ ℕ0)
12 simp3 1135 . . . . . . 7 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → 𝑅𝑉)
13 simp1 1133 . . . . . . . 8 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → ¬ 𝑁 = 1)
1413pm2.21d 121 . . . . . . 7 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → (𝑁 = 1 → Rel 𝑅))
1511, 12, 143jca 1125 . . . . . 6 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → (𝑁 ∈ ℕ0𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)))
16 relexprelg 14397 . . . . . 6 ((𝑁 ∈ ℕ0𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁))
17 relfld 6113 . . . . . 6 (Rel (𝑅𝑟𝑁) → (𝑅𝑟𝑁) = (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)))
1815, 16, 173syl 18 . . . . 5 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) = (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)))
19 elnn0 11896 . . . . . . 7 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
20 relexpnndm 14400 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ dom 𝑅)
21 relexpnnrn 14404 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑅𝑉) → ran (𝑅𝑟𝑁) ⊆ ran 𝑅)
22 unss12 4144 . . . . . . . . . 10 ((dom (𝑅𝑟𝑁) ⊆ dom 𝑅 ∧ ran (𝑅𝑟𝑁) ⊆ ran 𝑅) → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
2320, 21, 22syl2anc 587 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑅𝑉) → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
2423ex 416 . . . . . . . 8 (𝑁 ∈ ℕ → (𝑅𝑉 → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
25 simpl 486 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑅𝑉) → 𝑁 = 0)
2625oveq2d 7165 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
27 relexp0g 14381 . . . . . . . . . . . . . . 15 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
2827adantl 485 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
2926, 28eqtrd 2859 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
3029dmeqd 5761 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) = dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
31 dmresi 5908 . . . . . . . . . . . 12 dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
3230, 31syl6eq 2875 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅))
33 eqimss 4009 . . . . . . . . . . 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 5795 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑅𝑉) → ran (𝑅𝑟𝑁) = ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
36 rnresi 5930 . . . . . . . . . . . 12 ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
3735, 36syl6eq 2875 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑅𝑉) → ran (𝑅𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅))
38 eqimss 4009 . . . . . . . . . . 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 4148 . . . . . . . . 9 ((𝑁 = 0 ∧ 𝑅𝑉) → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
4140ex 416 . . . . . . . 8 (𝑁 = 0 → (𝑅𝑉 → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
4224, 41jaoi 854 . . . . . . 7 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑅𝑉 → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
4319, 42sylbi 220 . . . . . 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 3991 . . . 4 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
46 dmrnssfld 5828 . . . 4 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
4745, 46sstrdi 3965 . . 3 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) ⊆ 𝑅)
48473expib 1119 . 2 𝑁 = 1 → ((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) ⊆ 𝑅))
4910, 48pm2.61i 185 1 ((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) ⊆ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2115  cun 3917  wss 3919   cuni 4824   I cid 5446  dom cdm 5542  ran crn 5543  cres 5544  Rel wrel 5547  (class class class)co 7149  0cc0 10535  1c1 10536  cn 11634  0cn0 11894  𝑟crelexp 14379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-n0 11895  df-z 11979  df-uz 12241  df-seq 13374  df-relexp 14380
This theorem is referenced by:  relexpfldd  14409
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