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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 7936 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴) | |
2 | ot2ndg 7937 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵) | |
3 | ot3rdg 7938 | . . . 4 ⊢ (𝐶 ∈ 𝑍 → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶) | |
4 | 3 | 3ad2ant3 1136 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶) |
5 | 1, 2, 4 | 3jca 1129 | . 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 6848 | . . . 4 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (1st ‘(1st ‘𝑇)) = (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩))) | |
7 | 6 | eqeq1d 2739 | . . 3 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((1st ‘(1st ‘𝑇)) = 𝐴 ↔ (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)) |
8 | 2fveq3 6848 | . . . 4 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (2nd ‘(1st ‘𝑇)) = (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩))) | |
9 | 8 | eqeq1d 2739 | . . 3 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2nd ‘(1st ‘𝑇)) = 𝐵 ↔ (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)) |
10 | fveqeq2 6852 | . . 3 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2nd ‘𝑇) = 𝐶 ↔ (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)) | |
11 | 7, 9, 10 | 3anbi123d 1437 | . 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 246 | 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 1088 = wceq 1542 ∈ wcel 2107 ⟨cotp 4595 ‘cfv 6497 1st c1st 7920 2nd c2nd 7921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-ot 4596 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fv 6505 df-1st 7922 df-2nd 7923 |
This theorem is referenced by: (None) |
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