elrtrclrec - Mathbox for Richard Penner

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Theorem elrtrclrec 40032

Description: Membership in the indexed union of relation exponentiation over the natural numbers (including zero) is equivalent to the existence of at least one number such that the element is a member of that relationship power. (Contributed by RP, 2-Jun-2020.)

**Hypothesis**
Ref	Expression
rtrclrec.def	⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ₀ (𝑟↑_𝑟𝑛))

**Assertion**
Ref	Expression
elrtrclrec	⊢ (𝑅 ∈ 𝑉 → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ ℕ₀ 𝑋 ∈ (𝑅↑_𝑟𝑛)))

Distinct variable groups: 𝑛,𝑟,𝐶 𝑅,𝑛,𝑟 𝑛,𝑋 Allowed substitution hints: 𝑉(𝑛,𝑟) 𝑋(𝑟)

**Proof of Theorem elrtrclrec**
Step	Hyp	Ref	Expression
1		nn0ex 11906	. 2 ⊢ ℕ₀ ∈ V
2		rtrclrec.def	. . 3 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ₀ (𝑟↑_𝑟𝑛))
3	2	eliunov2 40030	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ ℕ₀ ∈ V) → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ ℕ₀ 𝑋 ∈ (𝑅↑_𝑟𝑛)))
4	1, 3	mpan2 689	1 ⊢ (𝑅 ∈ 𝑉 → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ ℕ₀ 𝑋 ∈ (𝑅↑_𝑟𝑛)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ∈ wcel 2113 ∃wrex 3142 Vcvv 3497 ∪ ciun 4922 ↦ cmpt 5149 ‘cfv 6358 (class class class)co 7159 ℕ₀cn0 11900 ↑_𝑟crelexp 14382

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-1cn 10598 ax-addcl 10600

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-nn 11642 df-n0 11901

This theorem is referenced by: (None)

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