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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 7692 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (1st ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐴) | |
2 | ot2ndg 7693 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐵) | |
3 | ot3rdg 7694 | . . . 4 ⊢ (𝐶 ∈ 𝑍 → (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶) | |
4 | 3 | 3ad2ant3 1127 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶) |
5 | 1, 2, 4 | 3jca 1120 | . 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 6668 | . . . 4 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → (1st ‘(1st ‘𝑇)) = (1st ‘(1st ‘〈𝐴, 𝐵, 𝐶〉))) | |
7 | 6 | eqeq1d 2820 | . . 3 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → ((1st ‘(1st ‘𝑇)) = 𝐴 ↔ (1st ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐴)) |
8 | 2fveq3 6668 | . . . 4 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → (2nd ‘(1st ‘𝑇)) = (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉))) | |
9 | 8 | eqeq1d 2820 | . . 3 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → ((2nd ‘(1st ‘𝑇)) = 𝐵 ↔ (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐵)) |
10 | fveqeq2 6672 | . . 3 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → ((2nd ‘𝑇) = 𝐶 ↔ (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶)) | |
11 | 7, 9, 10 | 3anbi123d 1427 | . 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 247 | 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 1079 = wceq 1528 ∈ wcel 2105 〈cotp 4565 ‘cfv 6348 1st c1st 7676 2nd c2nd 7677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-ot 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fv 6356 df-1st 7678 df-2nd 7679 |
This theorem is referenced by: (None) |
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