relexpreld - Metamath Proof Explorer

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Theorem relexpreld 14982

Description: The exponentiation of a relation is a relation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.)

**Hypotheses**
Ref	Expression
relexpreld.1	⊢ (𝜑 → Rel 𝑅)
relexpreld.2	⊢ (𝜑 → 𝑁 ∈ ℕ₀)

**Assertion**
Ref	Expression
relexpreld	⊢ (𝜑 → Rel (𝑅↑_𝑟𝑁))

**Proof of Theorem relexpreld**
Step	Hyp	Ref	Expression
1		relexpreld.2	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
2	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝑁 ∈ ℕ₀)
3		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝑅 ∈ V)
4		relexpreld.1	. . . . 5 ⊢ (𝜑 → Rel 𝑅)
5	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → Rel 𝑅)
6		relexprel 14981	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ V ∧ Rel 𝑅) → Rel (𝑅↑_𝑟𝑁))
7	2, 3, 5, 6	syl3anc 1373	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → Rel (𝑅↑_𝑟𝑁))
8	7	ex 412	. 2 ⊢ (𝜑 → (𝑅 ∈ V → Rel (𝑅↑_𝑟𝑁)))
9		rel0 5753	. . 3 ⊢ Rel ∅
10		reldmrelexp 14963	. . . . 5 ⊢ Rel dom ↑_𝑟
11	10	ovprc1 7408	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅↑_𝑟𝑁) = ∅)
12	11	releqd 5733	. . 3 ⊢ (¬ 𝑅 ∈ V → (Rel (𝑅↑_𝑟𝑁) ↔ Rel ∅))
13	9, 12	mpbiri 258	. 2 ⊢ (¬ 𝑅 ∈ V → Rel (𝑅↑_𝑟𝑁))
14	8, 13	pm2.61d1 180	1 ⊢ (𝜑 → Rel (𝑅↑_𝑟𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2109 Vcvv 3444 ∅c0 4292 Rel wrel 5636 (class class class)co 7369 ℕ₀cn0 12418 ↑_𝑟crelexp 14961

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 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-n0 12419 df-z 12506 df-uz 12770 df-seq 13943 df-relexp 14962

This theorem is referenced by: (None)

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