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Theorem relexpfld 13739
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 473 . . . . . . . 8 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → 𝑁 = 1)
21oveq2d 6631 . . . . . . 7 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟𝑁) = (𝑅𝑟1))
3 relexp1g 13716 . . . . . . . 8 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
43ad2antll 764 . . . . . . 7 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟1) = 𝑅)
52, 4eqtrd 2655 . . . . . 6 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟𝑁) = 𝑅)
65unieqd 4419 . . . . 5 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟𝑁) = 𝑅)
76unieqd 4419 . . . 4 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟𝑁) = 𝑅)
8 eqimss 3642 . . . 4 ( (𝑅𝑟𝑁) = 𝑅 (𝑅𝑟𝑁) ⊆ 𝑅)
97, 8syl 17 . . 3 ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0𝑅𝑉)) → (𝑅𝑟𝑁) ⊆ 𝑅)
109ex 450 . 2 (𝑁 = 1 → ((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) ⊆ 𝑅))
11 simp2 1060 . . . . . . 7 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → 𝑁 ∈ ℕ0)
12 simp3 1061 . . . . . . 7 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → 𝑅𝑉)
13 simp1 1059 . . . . . . . 8 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → ¬ 𝑁 = 1)
1413pm2.21d 118 . . . . . . 7 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → (𝑁 = 1 → Rel 𝑅))
1511, 12, 143jca 1240 . . . . . 6 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → (𝑁 ∈ ℕ0𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)))
16 relexprelg 13728 . . . . . 6 ((𝑁 ∈ ℕ0𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁))
17 relfld 5630 . . . . . 6 (Rel (𝑅𝑟𝑁) → (𝑅𝑟𝑁) = (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)))
1815, 16, 173syl 18 . . . . 5 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) = (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)))
19 elnn0 11254 . . . . . . 7 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
20 relexpnndm 13731 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ dom 𝑅)
21 relexpnnrn 13735 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑅𝑉) → ran (𝑅𝑟𝑁) ⊆ ran 𝑅)
22 unss12 3769 . . . . . . . . . 10 ((dom (𝑅𝑟𝑁) ⊆ dom 𝑅 ∧ ran (𝑅𝑟𝑁) ⊆ ran 𝑅) → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
2320, 21, 22syl2anc 692 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑅𝑉) → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
2423ex 450 . . . . . . . 8 (𝑁 ∈ ℕ → (𝑅𝑉 → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
25 simpl 473 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑅𝑉) → 𝑁 = 0)
2625oveq2d 6631 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
27 relexp0g 13712 . . . . . . . . . . . . . . 15 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
2827adantl 482 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
2926, 28eqtrd 2655 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
3029dmeqd 5296 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) = dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
31 dmresi 5426 . . . . . . . . . . . 12 dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
3230, 31syl6eq 2671 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅))
33 eqimss 3642 . . . . . . . . . . 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 5323 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑅𝑉) → ran (𝑅𝑟𝑁) = ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
36 rnresi 5448 . . . . . . . . . . . 12 ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
3735, 36syl6eq 2671 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑅𝑉) → ran (𝑅𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅))
38 eqimss 3642 . . . . . . . . . . 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 3773 . . . . . . . . 9 ((𝑁 = 0 ∧ 𝑅𝑉) → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
4140ex 450 . . . . . . . 8 (𝑁 = 0 → (𝑅𝑉 → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
4224, 41jaoi 394 . . . . . . 7 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑅𝑉 → (dom (𝑅𝑟𝑁) ∪ ran (𝑅𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
4319, 42sylbi 207 . . . . . 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 3624 . . . 4 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
46 dmrnssfld 5354 . . . 4 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
4745, 46syl6ss 3600 . . 3 ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) ⊆ 𝑅)
48473expib 1265 . 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  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  cun 3558  wss 3560   cuni 4409   I cid 4994  dom cdm 5084  ran crn 5085  cres 5086  Rel wrel 5089  (class class class)co 6615  0cc0 9896  1c1 9897  cn 10980  0cn0 11252  𝑟crelexp 13710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-n0 11253  df-z 11338  df-uz 11648  df-seq 12758  df-relexp 13711
This theorem is referenced by:  relexpfldd  13740
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