Theorem relexprelg 13626

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 11171	. . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		eqeq1 2614	. . . . . . . 8 ⊢ (𝑛 = 1 → (𝑛 = 1 ↔ 1 = 1))
3	2	imbi1d 330	. . . . . . 7 ⊢ (𝑛 = 1 → ((𝑛 = 1 → Rel 𝑅) ↔ (1 = 1 → Rel 𝑅)))
4	3	anbi2d 736	. . . . . 6 ⊢ (𝑛 = 1 → ((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅))))
5		oveq2 6557	. . . . . . 7 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
6	5	releqd 5126	. . . . . 6 ⊢ (𝑛 = 1 → (Rel (𝑅↑𝑟𝑛) ↔ Rel (𝑅↑𝑟1)))
7	4, 6	imbi12d 333	. . . . 5 ⊢ (𝑛 = 1 → (((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑛)) ↔ ((𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟1))))
8		eqeq1 2614	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑛 = 1 ↔ 𝑚 = 1))
9	8	imbi1d 330	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((𝑛 = 1 → Rel 𝑅) ↔ (𝑚 = 1 → Rel 𝑅)))
10	9	anbi2d 736	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅 ∈ 𝑉 ∧ (𝑚 = 1 → Rel 𝑅))))
11		oveq2 6557	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑚))
12	11	releqd 5126	. . . . . 6 ⊢ (𝑛 = 𝑚 → (Rel (𝑅↑𝑟𝑛) ↔ Rel (𝑅↑𝑟𝑚)))
13	10, 12	imbi12d 333	. . . . 5 ⊢ (𝑛 = 𝑚 → (((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑛)) ↔ ((𝑅 ∈ 𝑉 ∧ (𝑚 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑚))))
14		eqeq1 2614	. . . . . . . 8 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 = 1 ↔ (𝑚 + 1) = 1))
15	14	imbi1d 330	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → ((𝑛 = 1 → Rel 𝑅) ↔ ((𝑚 + 1) = 1 → Rel 𝑅)))
16	15	anbi2d 736	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → ((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅 ∈ 𝑉 ∧ ((𝑚 + 1) = 1 → Rel 𝑅))))
17		oveq2 6557	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
18	17	releqd 5126	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (Rel (𝑅↑𝑟𝑛) ↔ Rel (𝑅↑𝑟(𝑚 + 1))))
19	16, 18	imbi12d 333	. . . . 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 2614	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛 = 1 ↔ 𝑁 = 1))
21	20	imbi1d 330	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛 = 1 → Rel 𝑅) ↔ (𝑁 = 1 → Rel 𝑅)))
22	21	anbi2d 736	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅))))
23		oveq2 6557	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑁))
24	23	releqd 5126	. . . . . 6 ⊢ (𝑛 = 𝑁 → (Rel (𝑅↑𝑟𝑛) ↔ Rel (𝑅↑𝑟𝑁)))
25	22, 24	imbi12d 333	. . . . 5 ⊢ (𝑛 = 𝑁 → (((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑛)) ↔ ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁))))
26		eqid 2610	. . . . . . . 8 ⊢ 1 = 1
27		pm2.27 41	. . . . . . . 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 13614	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
31	30	adantr 480	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅)) → (𝑅↑𝑟1) = 𝑅)
32	31	releqd 5126	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅)) → (Rel (𝑅↑𝑟1) ↔ Rel 𝑅))
33	29, 32	mpbird 246	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟1))
34		relco 5550	. . . . . . . . 9 ⊢ Rel ((𝑅↑𝑟𝑚) ∘ 𝑅)
35		relexpsucnnr 13613	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
36	35	ancoms 468	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
37	36	releqd 5126	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (Rel (𝑅↑𝑟(𝑚 + 1)) ↔ Rel ((𝑅↑𝑟𝑚) ∘ 𝑅)))
38	34, 37	mpbiri 247	. . . . . . . 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 627	. . . . . 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 10915	. . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁)))
43		relexp0rel 13625	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → Rel (𝑅↑𝑟0))
44	43	adantl 481	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟0))
45		simpl 472	. . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0)
46	45	oveq2d 6565	. . . . . . . 8 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0))
47	46	releqd 5126	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (Rel (𝑅↑𝑟𝑁) ↔ Rel (𝑅↑𝑟0)))
48	44, 47	mpbird 246	. . . . . 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 627	. . . 4 ⊢ (𝑁 = 0 → ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁)))
51	42, 50	jaoi 393	. . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁)))
52	1, 51	sylbi 206	. 2 ⊢ (𝑁 ∈ ℕ0 → ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁)))
53	52	3impib 1254	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁))

Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∘ ccom 5042  Rel wrel 5043  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  ℕcn 10897  ℕ₀cn0 11169  ↑_𝑟crelexp 13608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-seq 12664  df-relexp 13609

This theorem is referenced by:  relexprel  13627  relexpfld  13637  relexpuzrel  13640