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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 5364 | . . . 4 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
2 | ot21st.3 | . . . 4 ⊢ 𝐶 ∈ V | |
3 | 1, 2 | op1std 7792 | . . 3 ⊢ (𝑋 = 〈〈𝐴, 𝐵〉, 𝐶〉 → (1st ‘𝑋) = 〈𝐴, 𝐵〉) |
4 | 3 | fveq2d 6742 | . 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 7791 | . 2 ⊢ (2nd ‘〈𝐴, 𝐵〉) = 𝐵 |
8 | 4, 7 | eqtrdi 2796 | 1 ⊢ (𝑋 = 〈〈𝐴, 𝐵〉, 𝐶〉 → (2nd ‘(1st ‘𝑋)) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 Vcvv 3423 〈cop 4563 ‘cfv 6400 1st c1st 7780 2nd c2nd 7781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5208 ax-nul 5215 ax-pr 5338 ax-un 7544 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3425 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4456 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4836 df-br 5070 df-opab 5132 df-mpt 5152 df-id 5471 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-iota 6358 df-fun 6402 df-fv 6408 df-1st 7782 df-2nd 7783 |
This theorem is referenced by: sbcoteq1a 33433 xpord3lem 33565 |
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