ot1st2nd - Mathbox for BTernaryTau

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Theorem ot1st2nd 35772

Description: Reconstruction of a member of a nested Cartesian product in terms of its ordered triple components. (Contributed by BTernaryTau, 9-Oct-2026.)

**Assertion**
Ref	Expression
ot1st2nd	⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) → 𝐴 = ⟨(1^st ‘(1^st ‘𝐴)), (2^nd ‘(1^st ‘𝐴)), (2^nd ‘𝐴)⟩)

**Proof of Theorem ot1st2nd**
Step	Hyp	Ref	Expression
1		1st2nd2 8044	. . 3 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
2		xp1st 8037	. . . . 5 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) → (1^st ‘𝐴) ∈ (𝐵 × 𝐶))
3		1st2nd2 8044	. . . . 5 ⊢ ((1^st ‘𝐴) ∈ (𝐵 × 𝐶) → (1^st ‘𝐴) = ⟨(1^st ‘(1^st ‘𝐴)), (2^nd ‘(1^st ‘𝐴))⟩)
4	2, 3	syl 18	. . . 4 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) → (1^st ‘𝐴) = ⟨(1^st ‘(1^st ‘𝐴)), (2^nd ‘(1^st ‘𝐴))⟩)
5	4	opeq1d 4845	. . 3 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ = ⟨⟨(1^st ‘(1^st ‘𝐴)), (2^nd ‘(1^st ‘𝐴))⟩, (2^nd ‘𝐴)⟩)
6	1, 5	eqtrd 2795	. 2 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) → 𝐴 = ⟨⟨(1^st ‘(1^st ‘𝐴)), (2^nd ‘(1^st ‘𝐴))⟩, (2^nd ‘𝐴)⟩)
7		df-ot 4858	. 2 ⊢ ⟨(1^st ‘(1^st ‘𝐴)), (2^nd ‘(1^st ‘𝐴)), (2^nd ‘𝐴)⟩ = ⟨⟨(1^st ‘(1^st ‘𝐴)), (2^nd ‘(1^st ‘𝐴))⟩, (2^nd ‘𝐴)⟩
8	6, 7	eqtr4di 2813	1 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) → 𝐴 = ⟨(1^st ‘(1^st ‘𝐴)), (2^nd ‘(1^st ‘𝐴)), (2^nd ‘𝐴)⟩)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1569 ∈ wcel 2144 ⟨cop 4834 ⟨cotp 4857 × cxp 5649 ‘cfv 6543 1^st c1st 8003 2^nd c2nd 8004

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pr 5391 ax-un 7755

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-ss 3897 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-sn 4585 df-pr 4587 df-op 4835 df-ot 4858 df-uni 4867 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-iota 6497 df-fun 6545 df-fv 6551 df-1st 8005 df-2nd 8006

This theorem is used by: dfk12 35963 dfk22 35964 dfk32 35965

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