ot1st2nd - Mathbox for BTernaryTau

	Mathbox for BTernaryTau		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > ot1st2nd		Structured version Visualization version GIF version

Theorem ot1st2nd 35836

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 8074	. . 3 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
2		xp1st 8067	. . . . 5 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) → (1^st ‘𝐴) ∈ (𝐵 × 𝐶))
3		1st2nd2 8074	. . . . 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 4851	. . 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 2798	. 2 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) → 𝐴 = ⟨⟨(1^st ‘(1^st ‘𝐴)), (2^nd ‘(1^st ‘𝐴))⟩, (2^nd ‘𝐴)⟩)
7		df-ot 4864	. 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 2816	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 1570 ∈ wcel 2147 ⟨cop 4840 ⟨cotp 4863 × cxp 5656 ‘cfv 6566 1^st c1st 8033 2^nd c2nd 8034

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pr 5398 ax-un 7783

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4841 df-ot 4864 df-uni 4873 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-iota 6520 df-fun 6568 df-fv 6574 df-1st 8035 df-2nd 8036

This theorem is used by: dfk12 36053 dfk22 36054 dfk32 36055

W3C validator