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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  ℕcn 11634  ℕ0cn0 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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