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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 7985 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴) | |
2 | ot2ndg 7986 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵) | |
3 | ot3rdg 7987 | . . . 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 6889 | . . . 4 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (1st ‘(1st ‘𝑇)) = (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩))) | |
7 | 6 | eqeq1d 2728 | . . 3 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((1st ‘(1st ‘𝑇)) = 𝐴 ↔ (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)) |
8 | 2fveq3 6889 | . . . 4 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (2nd ‘(1st ‘𝑇)) = (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩))) | |
9 | 8 | eqeq1d 2728 | . . 3 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2nd ‘(1st ‘𝑇)) = 𝐵 ↔ (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)) |
10 | fveqeq2 6893 | . . 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 | imbitrrid 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 1084 = wceq 1533 ∈ wcel 2098 ⟨cotp 4631 ‘cfv 6536 1st c1st 7969 2nd c2nd 7970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-ot 4632 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6488 df-fun 6538 df-fv 6544 df-1st 7971 df-2nd 7972 |
This theorem is referenced by: (None) |
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