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Mirrors > Home > MPE Home > Th. List > eleei | Structured version Visualization version GIF version |
Description: The forward direction of elee 26939. (Contributed by Scott Fenton, 1-Jul-2013.) |
Ref | Expression |
---|---|
eleei | ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝐴:(1...𝑁)⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleenn 26941 | . . 3 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ) | |
2 | elee 26939 | . . 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 270 | 1 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝐴:(1...𝑁)⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2112 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ℝcr 10693 1c1 10695 ℕcn 11795 ...cfz 13060 𝔼cee 26933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-map 8488 df-ee 26936 |
This theorem is referenced by: eedimeq 26943 fveere 26946 eqeefv 26948 |
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