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Theorem ot22ndd 33427
Description: Extract the second member of an ordered triple. Deduction version. (Contributed by Scott Fenton, 21-Aug-2024.)
Hypotheses
Ref Expression
ot21st.1 𝐴 ∈ V
ot21st.2 𝐵 ∈ V
ot21st.3 𝐶 ∈ V
Assertion
Ref Expression
ot22ndd (𝑋 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ → (2nd ‘(1st𝑋)) = 𝐵)

Proof of Theorem ot22ndd
StepHypRef Expression
1 opex 5364 . . . 4 𝐴, 𝐵⟩ ∈ V
2 ot21st.3 . . . 4 𝐶 ∈ V
31, 2op1std 7792 . . 3 (𝑋 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ → (1st𝑋) = ⟨𝐴, 𝐵⟩)
43fveq2d 6742 . 2 (𝑋 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ → (2nd ‘(1st𝑋)) = (2nd ‘⟨𝐴, 𝐵⟩))
5 ot21st.1 . . 3 𝐴 ∈ V
6 ot21st.2 . . 3 𝐵 ∈ V
75, 6op2nd 7791 . 2 (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵
84, 7eqtrdi 2796 1 (𝑋 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ → (2nd ‘(1st𝑋)) = 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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