Theorem relexprelg 15011

Description: The exponentiation of a class is a relation except when the exponent is one and the class is not a relation. (Contributed by RP, 23-May-2020.)

Assertion

Ref	Expression
relexprelg	⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁))

Proof of Theorem relexprelg

Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnn0 12451	. . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		eqeq1 2734	. . . . . . . 8 ⊢ (𝑛 = 1 → (𝑛 = 1 ↔ 1 = 1))
3	2	imbi1d 341	. . . . . . 7 ⊢ (𝑛 = 1 → ((𝑛 = 1 → Rel 𝑅) ↔ (1 = 1 → Rel 𝑅)))
4	3	anbi2d 630	. . . . . 6 ⊢ (𝑛 = 1 → ((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅))))
5		oveq2 7398	. . . . . . 7 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
6	5	releqd 5744	. . . . . 6 ⊢ (𝑛 = 1 → (Rel (𝑅↑𝑟𝑛) ↔ Rel (𝑅↑𝑟1)))
7	4, 6	imbi12d 344	. . . . 5 ⊢ (𝑛 = 1 → (((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑛)) ↔ ((𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟1))))
8		eqeq1 2734	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑛 = 1 ↔ 𝑚 = 1))
9	8	imbi1d 341	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((𝑛 = 1 → Rel 𝑅) ↔ (𝑚 = 1 → Rel 𝑅)))
10	9	anbi2d 630	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅 ∈ 𝑉 ∧ (𝑚 = 1 → Rel 𝑅))))
11		oveq2 7398	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑚))
12	11	releqd 5744	. . . . . 6 ⊢ (𝑛 = 𝑚 → (Rel (𝑅↑𝑟𝑛) ↔ Rel (𝑅↑𝑟𝑚)))
13	10, 12	imbi12d 344	. . . . 5 ⊢ (𝑛 = 𝑚 → (((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑛)) ↔ ((𝑅 ∈ 𝑉 ∧ (𝑚 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑚))))
14		eqeq1 2734	. . . . . . . 8 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 = 1 ↔ (𝑚 + 1) = 1))
15	14	imbi1d 341	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → ((𝑛 = 1 → Rel 𝑅) ↔ ((𝑚 + 1) = 1 → Rel 𝑅)))
16	15	anbi2d 630	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → ((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅 ∈ 𝑉 ∧ ((𝑚 + 1) = 1 → Rel 𝑅))))
17		oveq2 7398	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
18	17	releqd 5744	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (Rel (𝑅↑𝑟𝑛) ↔ Rel (𝑅↑𝑟(𝑚 + 1))))
19	16, 18	imbi12d 344	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑛)) ↔ ((𝑅 ∈ 𝑉 ∧ ((𝑚 + 1) = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟(𝑚 + 1)))))
20		eqeq1 2734	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛 = 1 ↔ 𝑁 = 1))
21	20	imbi1d 341	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛 = 1 → Rel 𝑅) ↔ (𝑁 = 1 → Rel 𝑅)))
22	21	anbi2d 630	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅))))
23		oveq2 7398	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑁))
24	23	releqd 5744	. . . . . 6 ⊢ (𝑛 = 𝑁 → (Rel (𝑅↑𝑟𝑛) ↔ Rel (𝑅↑𝑟𝑁)))
25	22, 24	imbi12d 344	. . . . 5 ⊢ (𝑛 = 𝑁 → (((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑛)) ↔ ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁))))
26		eqid 2730	. . . . . . . 8 ⊢ 1 = 1
27		pm2.27 42	. . . . . . . 8 ⊢ (1 = 1 → ((1 = 1 → Rel 𝑅) → Rel 𝑅))
28	26, 27	ax-mp 5	. . . . . . 7 ⊢ ((1 = 1 → Rel 𝑅) → Rel 𝑅)
29	28	adantl 481	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel 𝑅)
30		relexp1g 14999	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
31	30	adantr 480	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅)) → (𝑅↑𝑟1) = 𝑅)
32	31	releqd 5744	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅)) → (Rel (𝑅↑𝑟1) ↔ Rel 𝑅))
33	29, 32	mpbird 257	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟1))
34		relco 6082	. . . . . . . . 9 ⊢ Rel ((𝑅↑𝑟𝑚) ∘ 𝑅)
35		relexpsucnnr 14998	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
36	35	ancoms 458	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
37	36	releqd 5744	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (Rel (𝑅↑𝑟(𝑚 + 1)) ↔ Rel ((𝑅↑𝑟𝑚) ∘ 𝑅)))
38	34, 37	mpbiri 258	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟(𝑚 + 1)))
39	38	a1d 25	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (((𝑚 + 1) = 1 → Rel 𝑅) → Rel (𝑅↑𝑟(𝑚 + 1))))
40	39	expimpd 453	. . . . . 6 ⊢ (𝑚 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ ((𝑚 + 1) = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟(𝑚 + 1))))
41	40	a1d 25	. . . . 5 ⊢ (𝑚 ∈ ℕ → (((𝑅 ∈ 𝑉 ∧ (𝑚 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑚)) → ((𝑅 ∈ 𝑉 ∧ ((𝑚 + 1) = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟(𝑚 + 1)))))
42	7, 13, 19, 25, 33, 41	nnind 12211	. . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁)))
43		relexp0rel 15010	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → Rel (𝑅↑𝑟0))
44	43	adantl 481	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟0))
45		simpl 482	. . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0)
46	45	oveq2d 7406	. . . . . . . 8 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0))
47	46	releqd 5744	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (Rel (𝑅↑𝑟𝑁) ↔ Rel (𝑅↑𝑟0)))
48	44, 47	mpbird 257	. . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑁))
49	48	a1d 25	. . . . 5 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑁 = 1 → Rel 𝑅) → Rel (𝑅↑𝑟𝑁)))
50	49	expimpd 453	. . . 4 ⊢ (𝑁 = 0 → ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁)))
51	42, 50	jaoi 857	. . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁)))
52	1, 51	sylbi 217	. 2 ⊢ (𝑁 ∈ ℕ0 → ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁)))
53	52	3impib 1116	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ∘ ccom 5645  Rel wrel 5646  (class class class)co 7390  0cc0 11075  1c1 11076   + caddc 11078  ℕcn 12193  ℕ₀cn0 12449  ↑_𝑟crelexp 14992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-n0 12450  df-z 12537  df-uz 12801  df-seq 13974  df-relexp 14993

This theorem is referenced by:  relexprel  15012  relexpfld  15022  relexpuzrel  15025