Theorem relexpfld 14748

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 483	. . . . . . . 8 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉)) → 𝑁 = 1)
2	1	oveq2d 7284	. . . . . . 7 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉)) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟1))
3		relexp1g 14725	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
4	3	ad2antll 726	. . . . . . 7 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉)) → (𝑅↑𝑟1) = 𝑅)
5	2, 4	eqtrd 2778	. . . . . 6 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉)) → (𝑅↑𝑟𝑁) = 𝑅)
6	5	unieqd 4854	. . . . 5 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉)) → ∪ (𝑅↑𝑟𝑁) = ∪ 𝑅)
7	6	unieqd 4854	. . . 4 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉)) → ∪ ∪ (𝑅↑𝑟𝑁) = ∪ ∪ 𝑅)
8		eqimss 3977	. . . 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 413	. 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 14737	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁))
17		relfld 6172	. . . . . 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 12223	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
20		relexpnndm 14740	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅)
21		relexpnnrn 14744	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) ⊆ ran 𝑅)
22		unss12 4116	. . . . . . . . . 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 584	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (dom (𝑅↑𝑟𝑁) ∪ ran (𝑅↑𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
24	23	ex 413	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑅 ∈ 𝑉 → (dom (𝑅↑𝑟𝑁) ∪ ran (𝑅↑𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
25		simpl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0)
26	25	oveq2d 7284	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0))
27		relexp0g 14721	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
28	27	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
29	26, 28	eqtrd 2778	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
30	29	dmeqd 5808	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) = dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
31		dmresi 5955	. . . . . . . . . . . 12 ⊢ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
32	30, 31	eqtrdi 2794	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅))
33		eqimss 3977	. . . . . . . . . . 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 5841	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) = ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
36		rnresi 5977	. . . . . . . . . . . 12 ⊢ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
37	35, 36	eqtrdi 2794	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅))
38		eqimss 3977	. . . . . . . . . . 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 4120	. . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (dom (𝑅↑𝑟𝑁) ∪ ran (𝑅↑𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
41	40	ex 413	. . . . . . . 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 3959	. . . 4 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
46		dmrnssfld 5873	. . . 4 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅
47	45, 46	sstrdi 3933	. . 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 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ∪ cun 3885   ⊆ wss 3887  ∪ cuni 4840   I cid 5484  dom cdm 5585  ran crn 5586   ↾ cres 5587  Rel wrel 5590  (class class class)co 7268  0cc0 10859  1c1 10860  ℕcn 11961  ℕ₀cn0 12221  ↑_𝑟crelexp 14718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pow 5287  ax-pr 5351  ax-un 7579  ax-cnex 10915  ax-resscn 10916  ax-1cn 10917  ax-icn 10918  ax-addcl 10919  ax-addrcl 10920  ax-mulcl 10921  ax-mulrcl 10922  ax-mulcom 10923  ax-addass 10924  ax-mulass 10925  ax-distr 10926  ax-i2m1 10927  ax-1ne0 10928  ax-1rid 10929  ax-rnegex 10930  ax-rrecex 10931  ax-cnre 10932  ax-pre-lttri 10933  ax-pre-lttrn 10934  ax-pre-ltadd 10935  ax-pre-mulgt0 10936

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3432  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-iun 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5485  df-eprel 5491  df-po 5499  df-so 5500  df-fr 5540  df-we 5542  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-res 5597  df-ima 5598  df-pred 6196  df-ord 6263  df-on 6264  df-lim 6265  df-suc 6266  df-iota 6385  df-fun 6429  df-fn 6430  df-f 6431  df-f1 6432  df-fo 6433  df-f1o 6434  df-fv 6435  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-om 7704  df-2nd 7822  df-frecs 8085  df-wrecs 8116  df-recs 8190  df-rdg 8229  df-er 8486  df-en 8722  df-dom 8723  df-sdom 8724  df-pnf 10999  df-mnf 11000  df-xr 11001  df-ltxr 11002  df-le 11003  df-sub 11195  df-neg 11196  df-nn 11962  df-n0 12222  df-z 12308  df-uz 12571  df-seq 13710  df-relexp 14719

This theorem is referenced by:  relexpfldd  14749