Theorem relexpfld 14408

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 485	. . . . . . . 8 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉)) → 𝑁 = 1)
2	1	oveq2d 7172	. . . . . . 7 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉)) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟1))
3		relexp1g 14385	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
4	3	ad2antll 727	. . . . . . 7 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉)) → (𝑅↑𝑟1) = 𝑅)
5	2, 4	eqtrd 2856	. . . . . 6 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉)) → (𝑅↑𝑟𝑁) = 𝑅)
6	5	unieqd 4852	. . . . 5 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉)) → ∪ (𝑅↑𝑟𝑁) = ∪ 𝑅)
7	6	unieqd 4852	. . . 4 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉)) → ∪ ∪ (𝑅↑𝑟𝑁) = ∪ ∪ 𝑅)
8		eqimss 4023	. . . 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 415	. 2 ⊢ (𝑁 = 1 → ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅))
11		simp2 1133	. . . . . . 7 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℕ0)
12		simp3 1134	. . . . . . 7 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
13		simp1 1132	. . . . . . . 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 1124	. . . . . 6 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → (𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)))
16		relexprelg 14397	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁))
17		relfld 6126	. . . . . 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 11900	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
20		relexpnndm 14400	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅)
21		relexpnnrn 14404	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) ⊆ ran 𝑅)
22		unss12 4158	. . . . . . . . . 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 586	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (dom (𝑅↑𝑟𝑁) ∪ ran (𝑅↑𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
24	23	ex 415	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑅 ∈ 𝑉 → (dom (𝑅↑𝑟𝑁) ∪ ran (𝑅↑𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
25		simpl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0)
26	25	oveq2d 7172	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0))
27		relexp0g 14381	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
28	27	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
29	26, 28	eqtrd 2856	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
30	29	dmeqd 5774	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) = dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
31		dmresi 5921	. . . . . . . . . . . 12 ⊢ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
32	30, 31	syl6eq 2872	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅))
33		eqimss 4023	. . . . . . . . . . 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 5808	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) = ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
36		rnresi 5943	. . . . . . . . . . . 12 ⊢ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
37	35, 36	syl6eq 2872	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅))
38		eqimss 4023	. . . . . . . . . . 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 4162	. . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (dom (𝑅↑𝑟𝑁) ∪ ran (𝑅↑𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
41	40	ex 415	. . . . . . . 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 219	. . . . . 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 4005	. . . 4 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
46		dmrnssfld 5841	. . . 4 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅
47	45, 46	sstrdi 3979	. . 3 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅)
48	47	3expib 1118	. 2 ⊢ (¬ 𝑁 = 1 → ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅))
49	10, 48	pm2.61i 184	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ∪ cun 3934   ⊆ wss 3936  ∪ cuni 4838   I cid 5459  dom cdm 5555  ran crn 5556   ↾ cres 5557  Rel wrel 5560  (class class class)co 7156  0cc0 10537  1c1 10538  ℕcn 11638  ℕ₀cn0 11898  ↑_𝑟crelexp 14379

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-n0 11899  df-z 11983  df-uz 12245  df-seq 13371  df-relexp 14380

This theorem is referenced by:  relexpfldd  14409