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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 7073 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (1st ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐴) | |
2 | ot2ndg 7074 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐵) | |
3 | ot3rdg 7075 | . . . 4 ⊢ (𝐶 ∈ 𝑍 → (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶) | |
4 | 3 | 3ad2ant3 1077 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶) |
5 | 1, 2, 4 | 3jca 1235 | . 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 | fveq2 6103 | . . . . 5 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → (1st ‘𝑇) = (1st ‘〈𝐴, 𝐵, 𝐶〉)) | |
7 | 6 | fveq2d 6107 | . . . 4 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → (1st ‘(1st ‘𝑇)) = (1st ‘(1st ‘〈𝐴, 𝐵, 𝐶〉))) |
8 | 7 | eqeq1d 2612 | . . 3 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → ((1st ‘(1st ‘𝑇)) = 𝐴 ↔ (1st ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐴)) |
9 | 6 | fveq2d 6107 | . . . 4 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → (2nd ‘(1st ‘𝑇)) = (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉))) |
10 | 9 | eqeq1d 2612 | . . 3 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → ((2nd ‘(1st ‘𝑇)) = 𝐵 ↔ (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐵)) |
11 | fveq2 6103 | . . . 4 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → (2nd ‘𝑇) = (2nd ‘〈𝐴, 𝐵, 𝐶〉)) | |
12 | 11 | eqeq1d 2612 | . . 3 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → ((2nd ‘𝑇) = 𝐶 ↔ (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶)) |
13 | 8, 10, 12 | 3anbi123d 1391 | . 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))) |
14 | 5, 13 | syl5ibr 235 | 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 1031 = wceq 1475 ∈ wcel 1977 〈cotp 4133 ‘cfv 5804 1st c1st 7057 2nd c2nd 7058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-ot 4134 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-1st 7059 df-2nd 7060 |
This theorem is referenced by: el2wlkonot 26396 el2spthonot 26397 el2spthonot0 26398 |
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