Theorem ot21std 33203

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 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⟩ → (1st ‘(1st ‘𝑋)) = (1st ‘⟨𝐴, 𝐵⟩))
5		ot21st.1	. . 3 ⊢ 𝐴 ∈ V
6		ot21st.2	. . 3 ⊢ 𝐵 ∈ V
7	5, 6	op1st 7706	. 2 ⊢ (1st ‘⟨𝐴, 𝐵⟩) = 𝐴
8	4, 7	eqtrdi 2809	1 ⊢ (𝑋 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ → (1st ‘(1st ‘𝑋)) = 𝐴)

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ⟨cop 4531  ‘cfv 6339  1^st c1st 7696

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

This theorem is referenced by:  sbcoteq1a  33210  xpord3lem  33354