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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 7685 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (1st ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐴) | |
2 | ot2ndg 7686 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐵) | |
3 | ot3rdg 7687 | . . . 4 ⊢ (𝐶 ∈ 𝑍 → (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶) | |
4 | 3 | 3ad2ant3 1132 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶) |
5 | 1, 2, 4 | 3jca 1125 | . 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 6650 | . . . 4 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → (1st ‘(1st ‘𝑇)) = (1st ‘(1st ‘〈𝐴, 𝐵, 𝐶〉))) | |
7 | 6 | eqeq1d 2800 | . . 3 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → ((1st ‘(1st ‘𝑇)) = 𝐴 ↔ (1st ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐴)) |
8 | 2fveq3 6650 | . . . 4 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → (2nd ‘(1st ‘𝑇)) = (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉))) | |
9 | 8 | eqeq1d 2800 | . . 3 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → ((2nd ‘(1st ‘𝑇)) = 𝐵 ↔ (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐵)) |
10 | fveqeq2 6654 | . . 3 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → ((2nd ‘𝑇) = 𝐶 ↔ (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶)) | |
11 | 7, 9, 10 | 3anbi123d 1433 | . 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 249 | 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 1084 = wceq 1538 ∈ wcel 2111 〈cotp 4533 ‘cfv 6324 1st c1st 7669 2nd c2nd 7670 |
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 |
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-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-ot 4534 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-iota 6283 df-fun 6326 df-fv 6332 df-1st 7671 df-2nd 7672 |
This theorem is referenced by: (None) |
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