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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 7838 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (1st ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐴) | |
2 | ot2ndg 7839 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐵) | |
3 | ot3rdg 7840 | . . . 4 ⊢ (𝐶 ∈ 𝑍 → (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶) | |
4 | 3 | 3ad2ant3 1134 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶) |
5 | 1, 2, 4 | 3jca 1127 | . 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 6776 | . . . 4 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → (1st ‘(1st ‘𝑇)) = (1st ‘(1st ‘〈𝐴, 𝐵, 𝐶〉))) | |
7 | 6 | eqeq1d 2742 | . . 3 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → ((1st ‘(1st ‘𝑇)) = 𝐴 ↔ (1st ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐴)) |
8 | 2fveq3 6776 | . . . 4 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → (2nd ‘(1st ‘𝑇)) = (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉))) | |
9 | 8 | eqeq1d 2742 | . . 3 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → ((2nd ‘(1st ‘𝑇)) = 𝐵 ↔ (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐵)) |
10 | fveqeq2 6780 | . . 3 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → ((2nd ‘𝑇) = 𝐶 ↔ (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶)) | |
11 | 7, 9, 10 | 3anbi123d 1435 | . 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 245 | 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 1086 = wceq 1542 ∈ wcel 2110 〈cotp 4575 ‘cfv 6432 1st c1st 7822 2nd c2nd 7823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-ot 4576 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-iota 6390 df-fun 6434 df-fv 6440 df-1st 7824 df-2nd 7825 |
This theorem is referenced by: (None) |
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