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Mirrors > Home > MPE Home > Th. List > eqeefv | Structured version Visualization version GIF version |
Description: Two points are equal iff they agree in all dimensions. (Contributed by Scott Fenton, 10-Jun-2013.) |
Ref | Expression |
---|---|
eqeefv | ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (𝐴 = 𝐵 ↔ ∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (𝐵‘𝑖))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleei 26691 | . . 3 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝐴:(1...𝑁)⟶ℝ) | |
2 | 1 | ffnd 6488 | . 2 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝐴 Fn (1...𝑁)) |
3 | eleei 26691 | . . 3 ⊢ (𝐵 ∈ (𝔼‘𝑁) → 𝐵:(1...𝑁)⟶ℝ) | |
4 | 3 | ffnd 6488 | . 2 ⊢ (𝐵 ∈ (𝔼‘𝑁) → 𝐵 Fn (1...𝑁)) |
5 | eqfnfv 6779 | . 2 ⊢ ((𝐴 Fn (1...𝑁) ∧ 𝐵 Fn (1...𝑁)) → (𝐴 = 𝐵 ↔ ∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (𝐵‘𝑖))) | |
6 | 2, 4, 5 | syl2an 598 | 1 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (𝐴 = 𝐵 ↔ ∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (𝐵‘𝑖))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Fn wfn 6319 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 1c1 10527 ...cfz 12885 𝔼cee 26682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 df-ee 26685 |
This theorem is referenced by: eqeelen 26698 brbtwn2 26699 colinearalg 26704 axcgrid 26710 ax5seglem4 26726 ax5seglem5 26727 axbtwnid 26733 axeuclid 26757 axcontlem2 26759 axcontlem4 26761 axcontlem7 26764 |
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