Theorem relexpfld 14688

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 7271	. . . . . . 7 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉)) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟1))
3		relexp1g 14665	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
4	3	ad2antll 725	. . . . . . 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 4850	. . . . 5 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉)) → ∪ (𝑅↑𝑟𝑁) = ∪ 𝑅)
7	6	unieqd 4850	. . . 4 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉)) → ∪ ∪ (𝑅↑𝑟𝑁) = ∪ ∪ 𝑅)
8		eqimss 3973	. . . 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 1135	. . . . . . 7 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℕ0)
12		simp3 1136	. . . . . . 7 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
13		simp1 1134	. . . . . . . 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 1126	. . . . . 6 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)))
16		relexprelg 14677	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁))
17		relfld 6167	. . . . . 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 12165	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
20		relexpnndm 14680	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅)
21		relexpnnrn 14684	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) ⊆ ran 𝑅)
22		unss12 4112	. . . . . . . . . 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 7271	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0))
27		relexp0g 14661	. . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
30	29	dmeqd 5803	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) = dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
31		dmresi 5950	. . . . . . . . . . . 12 ⊢ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
32	30, 31	eqtrdi 2795	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅))
33		eqimss 3973	. . . . . . . . . . 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 5836	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) = ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
36		rnresi 5972	. . . . . . . . . . . 12 ⊢ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
37	35, 36	eqtrdi 2795	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅))
38		eqimss 3973	. . . . . . . . . . 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 4116	. . . . . . . . 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 853	. . . . . . 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 3955	. . . 4 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
46		dmrnssfld 5868	. . . 4 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅
47	45, 46	sstrdi 3929	. . 3 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅)
48	47	3expib 1120	. 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 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ∪ cun 3881   ⊆ wss 3883  ∪ cuni 4836   I cid 5479  dom cdm 5580  ran crn 5581   ↾ cres 5582  Rel wrel 5585  (class class class)co 7255  0cc0 10802  1c1 10803  ℕcn 11903  ℕ₀cn0 12163  ↑_𝑟crelexp 14658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-seq 13650  df-relexp 14659

This theorem is referenced by:  relexpfldd  14689