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Mirrors > Home > MPE Home > Th. List > eleei | Structured version Visualization version GIF version |
Description: The forward direction of elee 26386. (Contributed by Scott Fenton, 1-Jul-2013.) |
Ref | Expression |
---|---|
eleei | ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝐴:(1...𝑁)⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleenn 26388 | . . 3 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ) | |
2 | elee 26386 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴:(1...𝑁)⟶ℝ)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ (𝔼‘𝑁) → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴:(1...𝑁)⟶ℝ)) |
4 | 3 | ibi 259 | 1 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝐴:(1...𝑁)⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∈ wcel 2050 ⟶wf 6186 ‘cfv 6190 (class class class)co 6978 ℝcr 10336 1c1 10338 ℕcn 11441 ...cfz 12711 𝔼cee 26380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-sbc 3684 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-op 4449 df-uni 4714 df-br 4931 df-opab 4993 df-mpt 5010 df-id 5313 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-fv 6198 df-ov 6981 df-oprab 6982 df-mpo 6983 df-map 8210 df-ee 26383 |
This theorem is referenced by: eedimeq 26390 fveere 26393 eqeefv 26395 |
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