Theorem ot21std 33583

Description: Extract the first 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
ot21std	⊢ (𝑋 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ → (1st ‘(1st ‘𝑋)) = 𝐴)

Proof of Theorem ot21std

Step	Hyp	Ref	Expression
1		opex 5373	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		ot21st.3	. . . 4 ⊢ 𝐶 ∈ V
3	1, 2	op1std 7814	. . 3 ⊢ (𝑋 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ → (1st ‘𝑋) = ⟨𝐴, 𝐵⟩)
4	3	fveq2d 6760	. 2 ⊢ (𝑋 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ → (1st ‘(1st ‘𝑋)) = (1st ‘⟨𝐴, 𝐵⟩))
5		ot21st.1	. . 3 ⊢ 𝐴 ∈ V
6		ot21st.2	. . 3 ⊢ 𝐵 ∈ V
7	5, 6	op1st 7812	. 2 ⊢ (1st ‘⟨𝐴, 𝐵⟩) = 𝐴
8	4, 7	eqtrdi 2795	1 ⊢ (𝑋 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ → (1st ‘(1st ‘𝑋)) = 𝐴)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ⟨cop 4564  ‘cfv 6418  1^st c1st 7802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fv 6426  df-1st 7804

This theorem is referenced by:  sbcoteq1a  33590  xpord3lem  33722