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Theorem relexpfld 13637
 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 472 . . . . . . . 8 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → 𝑁 = 1)
21oveq2d 6565 . . . . . . 7 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟𝑁) = (𝑅𝑟1))
3 relexp1g 13614 . . . . . . . 8 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
43ad2antll 761 . . . . . . 7 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟1) = 𝑅)
52, 4eqtrd 2644 . . . . . 6 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟𝑁) = 𝑅)
65unieqd 4382 . . . . 5 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟𝑁) = 𝑅)
76unieqd 4382 . . . 4 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟𝑁) = 𝑅)
8 eqimss 3620 . . . 4 ( (𝑅𝑟𝑁) = 𝑅 (𝑅𝑟𝑁) ⊆ 𝑅)
97, 8syl 17 . . 3 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟𝑁) ⊆ 𝑅)
109ex 449 . 2 (𝑁 = 1 → ((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) ⊆ 𝑅))
11 simp2 1055 . . . . . . 7 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → 𝑁 ∈ ℕ0)
12 simp3 1056 . . . . . . 7 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → 𝑅𝑉)
13 simp1 1054 . . . . . . . 8 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → ¬ 𝑁 = 1)
1413pm2.21d 117 . . . . . . 7 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → (𝑁 = 1 → Rel 𝑅))
1511, 12, 143jca 1235 . . . . . 6 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → (𝑁 ∈ ℕ0𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)))
16 relexprelg 13626 . . . . . 6 ((𝑁 ∈ ℕ0𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁))
17 relfld 5578 . . . . . 6 (Rel (𝑅𝑟𝑁) → (𝑅𝑟𝑁) = (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)))
1815, 16, 173syl 18 . . . . 5 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) = (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)))
19 elnn0 11171 . . . . . . 7 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
20 relexpnndm 13629 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ dom 𝑅)
21 relexpnnrn 13633 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑅𝑉) → ran (𝑅𝑟𝑁) ⊆ ran 𝑅)
22 unss12 3747 . . . . . . . . . 10 ((dom (𝑅𝑟𝑁) ⊆ dom 𝑅 ∧ ran (𝑅𝑟𝑁) ⊆ ran 𝑅) → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
2320, 21, 22syl2anc 691 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑅𝑉) → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
2423ex 449 . . . . . . . 8 (𝑁 ∈ ℕ → (𝑅𝑉 → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
25 simpl 472 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑅𝑉) → 𝑁 = 0)
2625oveq2d 6565 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
27 relexp0g 13610 . . . . . . . . . . . . . . 15 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
2827adantl 481 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
2926, 28eqtrd 2644 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
3029dmeqd 5248 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) = dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
31 dmresi 5376 . . . . . . . . . . . 12 dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
3230, 31syl6eq 2660 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅))
33 eqimss 3620 . . . . . . . . . . 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 5274 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑅𝑉) → ran (𝑅𝑟𝑁) = ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
36 rnresi 5398 . . . . . . . . . . . 12 ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
3735, 36syl6eq 2660 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑅𝑉) → ran (𝑅𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅))
38 eqimss 3620 . . . . . . . . . . 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 3751 . . . . . . . . 9 ((𝑁 = 0 ∧ 𝑅𝑉) → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
4140ex 449 . . . . . . . 8 (𝑁 = 0 → (𝑅𝑉 → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
4224, 41jaoi 393 . . . . . . 7 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑅𝑉 → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
4319, 42sylbi 206 . . . . . 6 (𝑁 ∈ ℕ0 → (𝑅𝑉 → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
4411, 12, 43sylc 63 . . . . 5 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
4518, 44eqsstrd 3602 . . . 4 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
46 dmrnssfld 5305 . . . 4 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
4745, 46syl6ss 3580 . . 3 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) ⊆ 𝑅)
48473expib 1260 . 2 𝑁 = 1 → ((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) ⊆ 𝑅))
4910, 48pm2.61i 175 1 ((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) ⊆ 𝑅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∪ cun 3538   ⊆ wss 3540  ∪ cuni 4372   I cid 4948  dom cdm 5038  ran crn 5039   ↾ cres 5040  Rel wrel 5043  (class class class)co 6549  0cc0 9815  1c1 9816  ℕcn 10897  ℕ0cn0 11169  ↑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:  relexpfldd  13638
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