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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 28831 | . . 3 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝐴:(1...𝑁)⟶ℝ) | |
2 | 1 | ffnd 6729 | . 2 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝐴 Fn (1...𝑁)) |
3 | eleei 28831 | . . 3 ⊢ (𝐵 ∈ (𝔼‘𝑁) → 𝐵:(1...𝑁)⟶ℝ) | |
4 | 3 | ffnd 6729 | . 2 ⊢ (𝐵 ∈ (𝔼‘𝑁) → 𝐵 Fn (1...𝑁)) |
5 | eqfnfv 7044 | . 2 ⊢ ((𝐴 Fn (1...𝑁) ∧ 𝐵 Fn (1...𝑁)) → (𝐴 = 𝐵 ↔ ∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (𝐵‘𝑖))) | |
6 | 2, 4, 5 | syl2an 594 | 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 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 Fn wfn 6549 ‘cfv 6554 (class class class)co 7424 ℝcr 11157 1c1 11159 ...cfz 13538 𝔼cee 28822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-map 8857 df-ee 28825 |
This theorem is referenced by: eqeelen 28838 brbtwn2 28839 colinearalg 28844 axcgrid 28850 ax5seglem4 28866 ax5seglem5 28867 axbtwnid 28873 axeuclid 28897 axcontlem2 28899 axcontlem4 28901 axcontlem7 28904 |
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