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Mirrors > Home > MPE Home > Th. List > Mathboxes > ot22ndd | Structured version Visualization version GIF version |
Description: Extract the second member of an ordered triple. Deduction version. (Contributed by Scott Fenton, 21-Aug-2024.) |
Ref | Expression |
---|---|
ot21st.1 | ⊢ 𝐴 ∈ V |
ot21st.2 | ⊢ 𝐵 ∈ V |
ot21st.3 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
ot22ndd | ⊢ (𝑋 = 〈〈𝐴, 𝐵〉, 𝐶〉 → (2nd ‘(1st ‘𝑋)) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 5327 | . . . 4 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
2 | ot21st.3 | . . . 4 ⊢ 𝐶 ∈ V | |
3 | 1, 2 | op1std 7708 | . . 3 ⊢ (𝑋 = 〈〈𝐴, 𝐵〉, 𝐶〉 → (1st ‘𝑋) = 〈𝐴, 𝐵〉) |
4 | 3 | fveq2d 6666 | . 2 ⊢ (𝑋 = 〈〈𝐴, 𝐵〉, 𝐶〉 → (2nd ‘(1st ‘𝑋)) = (2nd ‘〈𝐴, 𝐵〉)) |
5 | ot21st.1 | . . 3 ⊢ 𝐴 ∈ V | |
6 | ot21st.2 | . . 3 ⊢ 𝐵 ∈ V | |
7 | 5, 6 | op2nd 7707 | . 2 ⊢ (2nd ‘〈𝐴, 𝐵〉) = 𝐵 |
8 | 4, 7 | eqtrdi 2809 | 1 ⊢ (𝑋 = 〈〈𝐴, 𝐵〉, 𝐶〉 → (2nd ‘(1st ‘𝑋)) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3409 〈cop 4531 ‘cfv 6339 1st c1st 7696 2nd c2nd 7697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-sep 5172 ax-nul 5179 ax-pr 5301 ax-un 7464 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-rab 3079 df-v 3411 df-sbc 3699 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-br 5036 df-opab 5098 df-mpt 5116 df-id 5433 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-iota 6298 df-fun 6341 df-fv 6347 df-1st 7698 df-2nd 7699 |
This theorem is referenced by: sbcoteq1a 33210 xpord3lem 33354 |
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