Theorem relexprelg 14989

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 12475	. . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		eqeq1 2730	. . . . . . . 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 628	. . . . . 6 ⊢ (𝑛 = 1 → ((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅))))
5		oveq2 7412	. . . . . . 7 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
6	5	releqd 5771	. . . . . 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 2730	. . . . . . . 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 628	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅 ∈ 𝑉 ∧ (𝑚 = 1 → Rel 𝑅))))
11		oveq2 7412	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑚))
12	11	releqd 5771	. . . . . 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 2730	. . . . . . . 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 628	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → ((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅 ∈ 𝑉 ∧ ((𝑚 + 1) = 1 → Rel 𝑅))))
17		oveq2 7412	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
18	17	releqd 5771	. . . . . 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 2730	. . . . . . . 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 628	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅))))
23		oveq2 7412	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑁))
24	23	releqd 5771	. . . . . 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 2726	. . . . . . . 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 14977	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
31	30	adantr 480	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅)) → (𝑅↑𝑟1) = 𝑅)
32	31	releqd 5771	. . . . . 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 6100	. . . . . . . . 9 ⊢ Rel ((𝑅↑𝑟𝑚) ∘ 𝑅)
35		relexpsucnnr 14976	. . . . . . . . . . 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 5771	. . . . . . . . 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 12231	. . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁)))
43		relexp0rel 14988	. . . . . . . 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 7420	. . . . . . . 8 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0))
47	46	releqd 5771	. . . . . . 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 854	. . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁)))
52	1, 51	sylbi 216	. 2 ⊢ (𝑁 ∈ ℕ0 → ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁)))
53	52	3impib 1113	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ∘ ccom 5673  Rel wrel 5674  (class class class)co 7404  0cc0 11109  1c1 11110   + caddc 11112  ℕcn 12213  ℕ₀cn0 12473  ↑_𝑟crelexp 14970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-n0 12474  df-z 12560  df-uz 12824  df-seq 13970  df-relexp 14971

This theorem is referenced by:  relexprel  14990  relexpfld  15000  relexpuzrel  15003