Theorem ot21std 33680

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 5379	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		ot21st.3	. . . 4 ⊢ 𝐶 ∈ V
3	1, 2	op1std 7841	. . 3 ⊢ (𝑋 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ → (1st ‘𝑋) = ⟨𝐴, 𝐵⟩)
4	3	fveq2d 6778	. 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 7839	. 2 ⊢ (1st ‘⟨𝐴, 𝐵⟩) = 𝐴
8	4, 7	eqtrdi 2794	1 ⊢ (𝑋 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ → (1st ‘(1st ‘𝑋)) = 𝐴)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ⟨cop 4567  ‘cfv 6433  1^st c1st 7829

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fv 6441  df-1st 7831

This theorem is referenced by:  sbcoteq1a  33687  xpord3lem  33795