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Mirrors > Home > MPE Home > Th. List > oteqimp | Structured version Visualization version GIF version |
Description: The components of an ordered triple. (Contributed by Alexander van der Vekens, 2-Mar-2018.) |
Ref | Expression |
---|---|
oteqimp | ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → ((1st ‘(1st ‘𝑇)) = 𝐴 ∧ (2nd ‘(1st ‘𝑇)) = 𝐵 ∧ (2nd ‘𝑇) = 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ot1stg 7697 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (1st ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐴) | |
2 | ot2ndg 7698 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐵) | |
3 | ot3rdg 7699 | . . . 4 ⊢ (𝐶 ∈ 𝑍 → (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶) | |
4 | 3 | 3ad2ant3 1131 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶) |
5 | 1, 2, 4 | 3jca 1124 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → ((1st ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐴 ∧ (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐵 ∧ (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶)) |
6 | 2fveq3 6669 | . . . 4 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → (1st ‘(1st ‘𝑇)) = (1st ‘(1st ‘〈𝐴, 𝐵, 𝐶〉))) | |
7 | 6 | eqeq1d 2823 | . . 3 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → ((1st ‘(1st ‘𝑇)) = 𝐴 ↔ (1st ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐴)) |
8 | 2fveq3 6669 | . . . 4 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → (2nd ‘(1st ‘𝑇)) = (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉))) | |
9 | 8 | eqeq1d 2823 | . . 3 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → ((2nd ‘(1st ‘𝑇)) = 𝐵 ↔ (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐵)) |
10 | fveqeq2 6673 | . . 3 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → ((2nd ‘𝑇) = 𝐶 ↔ (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶)) | |
11 | 7, 9, 10 | 3anbi123d 1432 | . 2 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → (((1st ‘(1st ‘𝑇)) = 𝐴 ∧ (2nd ‘(1st ‘𝑇)) = 𝐵 ∧ (2nd ‘𝑇) = 𝐶) ↔ ((1st ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐴 ∧ (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐵 ∧ (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶))) |
12 | 5, 11 | syl5ibr 248 | 1 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → ((1st ‘(1st ‘𝑇)) = 𝐴 ∧ (2nd ‘(1st ‘𝑇)) = 𝐵 ∧ (2nd ‘𝑇) = 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 〈cotp 4568 ‘cfv 6349 1st c1st 7681 2nd c2nd 7682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-ot 4569 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fv 6357 df-1st 7683 df-2nd 7684 |
This theorem is referenced by: (None) |
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