relexpreld - Metamath Proof Explorer

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Theorem relexpreld 15012

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 15011	. . . 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 5764	. . 3 ⊢ Rel ∅
10		reldmrelexp 14993	. . . . 5 ⊢ Rel dom ↑_𝑟
11	10	ovprc1 7428	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅↑_𝑟𝑁) = ∅)
12	11	releqd 5743	. . 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 3450 ∅c0 4298 Rel wrel 5645 (class class class)co 7389 ℕ₀cn0 12448 ↑_𝑟crelexp 14991

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 2702 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-om 7845 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-er 8673 df-en 8921 df-dom 8922 df-sdom 8923 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-nn 12188 df-n0 12449 df-z 12536 df-uz 12800 df-seq 13973 df-relexp 14992

This theorem is referenced by: (None)

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