relexpfld - Metamath Proof Explorer

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Theorem relexpfld 14984

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 7384	. . . . . . 7 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (𝑅↑_𝑟𝑁) = (𝑅↑_𝑟1))
3		relexp1g 14961	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑_𝑟1) = 𝑅)
4	3	ad2antll 730	. . . . . . 7 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (𝑅↑_𝑟1) = 𝑅)
5	2, 4	eqtrd 2772	. . . . . 6 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (𝑅↑_𝑟𝑁) = 𝑅)
6	5	unieqd 4878	. . . . 5 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ (𝑅↑_𝑟𝑁) = ∪ 𝑅)
7	6	unieqd 4878	. . . 4 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ ∪ (𝑅↑_𝑟𝑁) = ∪ ∪ 𝑅)
8		eqimss 3994	. . . 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 1138	. . . . . . 7 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℕ₀)
12		simp3 1139	. . . . . . 7 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
13		simp1 1137	. . . . . . . 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 1129	. . . . . 6 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → (𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)))
16		relexprelg 14973	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑_𝑟𝑁))
17		relfld 6241	. . . . . 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 12415	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
20		relexpnndm 14976	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑_𝑟𝑁) ⊆ dom 𝑅)
21		relexpnnrn 14980	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑_𝑟𝑁) ⊆ ran 𝑅)
22		unss12 4142	. . . . . . . . . 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 585	. . . . . . . . 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 7384	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑_𝑟𝑁) = (𝑅↑_𝑟0))
27		relexp0g 14957	. . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑_𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
30	29	dmeqd 5862	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑_𝑟𝑁) = dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
31		dmresi 6019	. . . . . . . . . . . 12 ⊢ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
32	30, 31	eqtrdi 2788	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑_𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅))
33		eqimss 3994	. . . . . . . . . . 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 5895	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑_𝑟𝑁) = ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
36		rnresi 6042	. . . . . . . . . . . 12 ⊢ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
37	35, 36	eqtrdi 2788	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑_𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅))
38		eqimss 3994	. . . . . . . . . . 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 4146	. . . . . . . . 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 858	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑅 ∈ 𝑉 → (dom (𝑅↑_𝑟𝑁) ∪ ran (𝑅↑_𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
43	19, 42	sylbi 217	. . . . . 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 3970	. . . 4 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑_𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
46		dmrnssfld 5931	. . . 4 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅
47	45, 46	sstrdi 3948	. . 3 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑_𝑟𝑁) ⊆ ∪ ∪ 𝑅)
48	47	3expib 1123	. 2 ⊢ (¬ 𝑁 = 1 → ((𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑_𝑟𝑁) ⊆ ∪ ∪ 𝑅))
49	10, 48	pm2.61i 182	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑_𝑟𝑁) ⊆ ∪ ∪ 𝑅)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∪ cun 3901 ⊆ wss 3903 ∪ cuni 4865 I cid 5526 dom cdm 5632 ran crn 5633 ↾ cres 5634 Rel wrel 5637 (class class class)co 7368 0cc0 11038 1c1 11039 ℕcn 12157 ℕ₀cn0 12413 ↑_𝑟crelexp 14954

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-n0 12414 df-z 12501 df-uz 12764 df-seq 13937 df-relexp 14955

This theorem is referenced by: relexpfldd 14985

W3C validator