Theorem relexpfld 15001

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 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅)

Proof of Theorem relexpfld

Step	Hyp	Ref	Expression
1		simpl 482	. . . . . . . 8 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉)) → 𝑁 = 1)
2	1	oveq2d 7428	. . . . . . 7 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉)) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟1))
3		relexp1g 14978	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
4	3	ad2antll 726	. . . . . . 7 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉)) → (𝑅↑𝑟1) = 𝑅)
5	2, 4	eqtrd 2771	. . . . . 6 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉)) → (𝑅↑𝑟𝑁) = 𝑅)
6	5	unieqd 4922	. . . . 5 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉)) → ∪ (𝑅↑𝑟𝑁) = ∪ 𝑅)
7	6	unieqd 4922	. . . 4 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉)) → ∪ ∪ (𝑅↑𝑟𝑁) = ∪ ∪ 𝑅)
8		eqimss 4040	. . . 4 ⊢ (∪ ∪ (𝑅↑𝑟𝑁) = ∪ ∪ 𝑅 → ∪ ∪ (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅)
9	7, 8	syl 17	. . 3 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉)) → ∪ ∪ (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅)
10	9	ex 412	. 2 ⊢ (𝑁 = 1 → ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅))
11		simp2 1136	. . . . . . 7 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℕ0)
12		simp3 1137	. . . . . . 7 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
13		simp1 1135	. . . . . . . 8 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ¬ 𝑁 = 1)
14	13	pm2.21d 121	. . . . . . 7 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → (𝑁 = 1 → Rel 𝑅))
15	11, 12, 14	3jca 1127	. . . . . 6 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)))
16		relexprelg 14990	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁))
17		relfld 6274	. . . . . 6 ⊢ (Rel (𝑅↑𝑟𝑁) → ∪ ∪ (𝑅↑𝑟𝑁) = (dom (𝑅↑𝑟𝑁) ∪ ran (𝑅↑𝑟𝑁)))
18	15, 16, 17	3syl 18	. . . . 5 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑𝑟𝑁) = (dom (𝑅↑𝑟𝑁) ∪ ran (𝑅↑𝑟𝑁)))
19		elnn0 12479	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
20		relexpnndm 14993	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅)
21		relexpnnrn 14997	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) ⊆ ran 𝑅)
22		unss12 4182	. . . . . . . . . 10 ⊢ ((dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅 ∧ ran (𝑅↑𝑟𝑁) ⊆ ran 𝑅) → (dom (𝑅↑𝑟𝑁) ∪ ran (𝑅↑𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
23	20, 21, 22	syl2anc 583	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (dom (𝑅↑𝑟𝑁) ∪ ran (𝑅↑𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
24	23	ex 412	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑅 ∈ 𝑉 → (dom (𝑅↑𝑟𝑁) ∪ ran (𝑅↑𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
25		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0)
26	25	oveq2d 7428	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0))
27		relexp0g 14974	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
28	27	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
29	26, 28	eqtrd 2771	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
30	29	dmeqd 5905	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) = dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
31		dmresi 6051	. . . . . . . . . . . 12 ⊢ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
32	30, 31	eqtrdi 2787	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅))
33		eqimss 4040	. . . . . . . . . . 11 ⊢ (dom (𝑅↑𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅) → dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
34	32, 33	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
35	29	rneqd 5937	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) = ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
36		rnresi 6074	. . . . . . . . . . . 12 ⊢ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
37	35, 36	eqtrdi 2787	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅))
38		eqimss 4040	. . . . . . . . . . 11 ⊢ (ran (𝑅↑𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅) → ran (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
39	37, 38	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
40	34, 39	unssd 4186	. . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (dom (𝑅↑𝑟𝑁) ∪ ran (𝑅↑𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
41	40	ex 412	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑅 ∈ 𝑉 → (dom (𝑅↑𝑟𝑁) ∪ ran (𝑅↑𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
42	24, 41	jaoi 854	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑅 ∈ 𝑉 → (dom (𝑅↑𝑟𝑁) ∪ ran (𝑅↑𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
43	19, 42	sylbi 216	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑅 ∈ 𝑉 → (dom (𝑅↑𝑟𝑁) ∪ ran (𝑅↑𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
44	11, 12, 43	sylc 65	. . . . 5 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → (dom (𝑅↑𝑟𝑁) ∪ ran (𝑅↑𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
45	18, 44	eqsstrd 4020	. . . 4 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
46		dmrnssfld 5969	. . . 4 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅
47	45, 46	sstrdi 3994	. . 3 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅)
48	47	3expib 1121	. 2 ⊢ (¬ 𝑁 = 1 → ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅))
49	10, 48	pm2.61i 182	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 844   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ∪ cun 3946   ⊆ wss 3948  ∪ cuni 4908   I cid 5573  dom cdm 5676  ran crn 5677   ↾ cres 5678  Rel wrel 5681  (class class class)co 7412  0cc0 11114  1c1 11115  ℕcn 12217  ℕ₀cn0 12477  ↑_𝑟crelexp 14971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-n0 12478  df-z 12564  df-uz 12828  df-seq 13972  df-relexp 14972

This theorem is referenced by:  relexpfldd  15002