Theorem relexprelg 15065

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 12497	. . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		eqeq1 2769	. . . . . . . 8 ⊢ (𝑛 = 1 → (𝑛 = 1 ↔ 1 = 1))
3	2	imbi1d 344	. . . . . . 7 ⊢ (𝑛 = 1 → ((𝑛 = 1 → Rel 𝑅) ↔ (1 = 1 → Rel 𝑅)))
4	3	anbi2d 641	. . . . . 6 ⊢ (𝑛 = 1 → ((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅))))
5		oveq2 7408	. . . . . . 7 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
6	5	releqd 5756	. . . . . 6 ⊢ (𝑛 = 1 → (Rel (𝑅↑𝑟𝑛) ↔ Rel (𝑅↑𝑟1)))
7	4, 6	imbi12d 347	. . . . 5 ⊢ (𝑛 = 1 → (((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑛)) ↔ ((𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟1))))
8		eqeq1 2769	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑛 = 1 ↔ 𝑚 = 1))
9	8	imbi1d 344	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((𝑛 = 1 → Rel 𝑅) ↔ (𝑚 = 1 → Rel 𝑅)))
10	9	anbi2d 641	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅 ∈ 𝑉 ∧ (𝑚 = 1 → Rel 𝑅))))
11		oveq2 7408	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑚))
12	11	releqd 5756	. . . . . 6 ⊢ (𝑛 = 𝑚 → (Rel (𝑅↑𝑟𝑛) ↔ Rel (𝑅↑𝑟𝑚)))
13	10, 12	imbi12d 347	. . . . 5 ⊢ (𝑛 = 𝑚 → (((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑛)) ↔ ((𝑅 ∈ 𝑉 ∧ (𝑚 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑚))))
14		eqeq1 2769	. . . . . . . 8 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 = 1 ↔ (𝑚 + 1) = 1))
15	14	imbi1d 344	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → ((𝑛 = 1 → Rel 𝑅) ↔ ((𝑚 + 1) = 1 → Rel 𝑅)))
16	15	anbi2d 641	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → ((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅 ∈ 𝑉 ∧ ((𝑚 + 1) = 1 → Rel 𝑅))))
17		oveq2 7408	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
18	17	releqd 5756	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (Rel (𝑅↑𝑟𝑛) ↔ Rel (𝑅↑𝑟(𝑚 + 1))))
19	16, 18	imbi12d 347	. . . . 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 2769	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛 = 1 ↔ 𝑁 = 1))
21	20	imbi1d 344	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛 = 1 → Rel 𝑅) ↔ (𝑁 = 1 → Rel 𝑅)))
22	21	anbi2d 641	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅))))
23		oveq2 7408	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑁))
24	23	releqd 5756	. . . . . 6 ⊢ (𝑛 = 𝑁 → (Rel (𝑅↑𝑟𝑛) ↔ Rel (𝑅↑𝑟𝑁)))
25	22, 24	imbi12d 347	. . . . 5 ⊢ (𝑛 = 𝑁 → (((𝑅 ∈ 𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑛)) ↔ ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁))))
26		eqid 2765	. . . . . . . 8 ⊢ 1 = 1
27		pm2.27 43	. . . . . . . 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 486	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel 𝑅)
30		relexp1g 15053	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
31	30	adantr 485	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅)) → (𝑅↑𝑟1) = 𝑅)
32	31	releqd 5756	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅)) → (Rel (𝑅↑𝑟1) ↔ Rel 𝑅))
33	29, 32	mpbird 260	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟1))
34		relco 6101	. . . . . . . . 9 ⊢ Rel ((𝑅↑𝑟𝑚) ∘ 𝑅)
35		relexpsucnnr 15052	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
36	35	ancoms 463	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
37	36	releqd 5756	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (Rel (𝑅↑𝑟(𝑚 + 1)) ↔ Rel ((𝑅↑𝑟𝑚) ∘ 𝑅)))
38	34, 37	mpbiri 261	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟(𝑚 + 1)))
39	38	a1d 26	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (((𝑚 + 1) = 1 → Rel 𝑅) → Rel (𝑅↑𝑟(𝑚 + 1))))
40	39	expimpd 458	. . . . . 6 ⊢ (𝑚 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ ((𝑚 + 1) = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟(𝑚 + 1))))
41	40	a1d 26	. . . . 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 12242	. . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁)))
43		relexp0rel 15064	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → Rel (𝑅↑𝑟0))
44	43	adantl 486	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟0))
45		simpl 487	. . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0)
46	45	oveq2d 7416	. . . . . . . 8 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0))
47	46	releqd 5756	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (Rel (𝑅↑𝑟𝑁) ↔ Rel (𝑅↑𝑟0)))
48	44, 47	mpbird 260	. . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑁))
49	48	a1d 26	. . . . 5 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑁 = 1 → Rel 𝑅) → Rel (𝑅↑𝑟𝑁)))
50	49	expimpd 458	. . . 4 ⊢ (𝑁 = 0 → ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁)))
51	42, 50	jaoi 870	. . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁)))
52	1, 51	sylbi 220	. 2 ⊢ (𝑁 ∈ ℕ0 → ((𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁)))
53	52	3impib 1132	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ∘ ccom 5656  Rel wrel 5657  (class class class)co 7400  0cc0 11088  1c1 11089   + caddc 11091  ℕcn 12224  ℕ₀cn0 12495  ↑_𝑟crelexp 15046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-n0 12496  df-z 12583  df-uz 12854  df-seq 14029  df-relexp 15047

This theorem is referenced by:  relexprel  15066  relexpfld  15076  relexpuzrel  15079