Theorem relexprelg 13778

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 11294	. . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		eqeq1 2626	. . . . . . . 8 ⊢ (𝑛 = 1 → (𝑛 = 1 ↔ 1 = 1))
3	2	imbi1d 331	. . . . . . 7 ⊢ (𝑛 = 1 → ((𝑛 = 1 → Rel 𝑅) ↔ (1 = 1 → Rel 𝑅)))
4	3	anbi2d 740	. . . . . 6 ⊢ (𝑛 = 1 → ((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅))))
5		oveq2 6658	. . . . . . 7 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
6	5	releqd 5203	. . . . . 6 ⊢ (𝑛 = 1 → (Rel (𝑅↑𝑟𝑛) ↔ Rel (𝑅↑𝑟1)))
7	4, 6	imbi12d 334	. . . . 5 ⊢ (𝑛 = 1 → (((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑛)) ↔ ((𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟1))))
8		eqeq1 2626	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑛 = 1 ↔ 𝑚 = 1))
9	8	imbi1d 331	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((𝑛 = 1 → Rel 𝑅) ↔ (𝑚 = 1 → Rel 𝑅)))
10	9	anbi2d 740	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅 ∈ 𝑉 ∧ (𝑚 = 1 → Rel 𝑅))))
11		oveq2 6658	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑚))
12	11	releqd 5203	. . . . . 6 ⊢ (𝑛 = 𝑚 → (Rel (𝑅↑𝑟𝑛) ↔ Rel (𝑅↑𝑟𝑚)))
13	10, 12	imbi12d 334	. . . . 5 ⊢ (𝑛 = 𝑚 → (((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑛)) ↔ ((𝑅 ∈ 𝑉 ∧ (𝑚 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑚))))
14		eqeq1 2626	. . . . . . . 8 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 = 1 ↔ (𝑚 + 1) = 1))
15	14	imbi1d 331	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → ((𝑛 = 1 → Rel 𝑅) ↔ ((𝑚 + 1) = 1 → Rel 𝑅)))
16	15	anbi2d 740	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → ((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅 ∈ 𝑉 ∧ ((𝑚 + 1) = 1 → Rel 𝑅))))
17		oveq2 6658	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
18	17	releqd 5203	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (Rel (𝑅↑𝑟𝑛) ↔ Rel (𝑅↑𝑟(𝑚 + 1))))
19	16, 18	imbi12d 334	. . . . 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 2626	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛 = 1 ↔ 𝑁 = 1))
21	20	imbi1d 331	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛 = 1 → Rel 𝑅) ↔ (𝑁 = 1 → Rel 𝑅)))
22	21	anbi2d 740	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅))))
23		oveq2 6658	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑁))
24	23	releqd 5203	. . . . . 6 ⊢ (𝑛 = 𝑁 → (Rel (𝑅↑𝑟𝑛) ↔ Rel (𝑅↑𝑟𝑁)))
25	22, 24	imbi12d 334	. . . . 5 ⊢ (𝑛 = 𝑁 → (((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑛)) ↔ ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁))))
26		eqid 2622	. . . . . . . 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 482	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel 𝑅)
30		relexp1g 13766	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
31	30	adantr 481	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅)) → (𝑅↑𝑟1) = 𝑅)
32	31	releqd 5203	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅)) → (Rel (𝑅↑𝑟1) ↔ Rel 𝑅))
33	29, 32	mpbird 247	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟1))
34		relco 5633	. . . . . . . . 9 ⊢ Rel ((𝑅↑𝑟𝑚) ∘ 𝑅)
35		relexpsucnnr 13765	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
36	35	ancoms 469	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
37	36	releqd 5203	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (Rel (𝑅↑𝑟(𝑚 + 1)) ↔ Rel ((𝑅↑𝑟𝑚) ∘ 𝑅)))
38	34, 37	mpbiri 248	. . . . . . . 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 629	. . . . . 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 11038	. . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁)))
43		relexp0rel 13777	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → Rel (𝑅↑𝑟0))
44	43	adantl 482	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟0))
45		simpl 473	. . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0)
46	45	oveq2d 6666	. . . . . . . 8 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0))
47	46	releqd 5203	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (Rel (𝑅↑𝑟𝑁) ↔ Rel (𝑅↑𝑟0)))
48	44, 47	mpbird 247	. . . . . 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 629	. . . 4 ⊢ (𝑁 = 0 → ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁)))
51	42, 50	jaoi 394	. . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁)))
52	1, 51	sylbi 207	. 2 ⊢ (𝑁 ∈ ℕ0 → ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁)))
53	52	3impib 1262	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁))

Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ∘ ccom 5118  Rel wrel 5119  (class class class)co 6650  0cc0 9936  1c1 9937   + caddc 9939  ℕcn 11020  ℕ₀cn0 11292  ↑_𝑟crelexp 13760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-relexp 13761

This theorem is referenced by:  relexprel  13779  relexpfld  13789  relexpuzrel  13792