mpst123 - Mathbox for Mario Carneiro

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Theorem mpst123 35527

Description: Decompose a pre-statement into a triple of values. (Contributed by Mario Carneiro, 18-Jul-2016.)

**Hypothesis**
Ref	Expression
mpstssv.p	⊢ 𝑃 = (mPreSt‘𝑇)

**Assertion**
Ref	Expression
mpst123	⊢ (𝑋 ∈ 𝑃 → 𝑋 = ⟨(1^st ‘(1^st ‘𝑋)), (2^nd ‘(1^st ‘𝑋)), (2^nd ‘𝑋)⟩)

**Proof of Theorem mpst123**
Step	Hyp	Ref	Expression
1		mpstssv.p	. . . 4 ⊢ 𝑃 = (mPreSt‘𝑇)
2	1	mpstssv 35526	. . 3 ⊢ 𝑃 ⊆ ((V × V) × V)
3	2	sseli 3942	. 2 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ ((V × V) × V))
4		1st2nd2 8007	. . . 4 ⊢ (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
5		xp1st 8000	. . . . . 6 ⊢ (𝑋 ∈ ((V × V) × V) → (1^st ‘𝑋) ∈ (V × V))
6		1st2nd2 8007	. . . . . 6 ⊢ ((1^st ‘𝑋) ∈ (V × V) → (1^st ‘𝑋) = ⟨(1^st ‘(1^st ‘𝑋)), (2^nd ‘(1^st ‘𝑋))⟩)
7	5, 6	syl 17	. . . . 5 ⊢ (𝑋 ∈ ((V × V) × V) → (1^st ‘𝑋) = ⟨(1^st ‘(1^st ‘𝑋)), (2^nd ‘(1^st ‘𝑋))⟩)
8	7	opeq1d 4843	. . . 4 ⊢ (𝑋 ∈ ((V × V) × V) → ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ = ⟨⟨(1^st ‘(1^st ‘𝑋)), (2^nd ‘(1^st ‘𝑋))⟩, (2^nd ‘𝑋)⟩)
9	4, 8	eqtrd 2764	. . 3 ⊢ (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨⟨(1^st ‘(1^st ‘𝑋)), (2^nd ‘(1^st ‘𝑋))⟩, (2^nd ‘𝑋)⟩)
10		df-ot 4598	. . 3 ⊢ ⟨(1^st ‘(1^st ‘𝑋)), (2^nd ‘(1^st ‘𝑋)), (2^nd ‘𝑋)⟩ = ⟨⟨(1^st ‘(1^st ‘𝑋)), (2^nd ‘(1^st ‘𝑋))⟩, (2^nd ‘𝑋)⟩
11	9, 10	eqtr4di 2782	. 2 ⊢ (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨(1^st ‘(1^st ‘𝑋)), (2^nd ‘(1^st ‘𝑋)), (2^nd ‘𝑋)⟩)
12	3, 11	syl 17	1 ⊢ (𝑋 ∈ 𝑃 → 𝑋 = ⟨(1^st ‘(1^st ‘𝑋)), (2^nd ‘(1^st ‘𝑋)), (2^nd ‘𝑋)⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 Vcvv 3447 ⟨cop 4595 ⟨cotp 4597 × cxp 5636 ‘cfv 6511 1^st c1st 7966 2^nd c2nd 7967 mPreStcmpst 35460

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-ot 4598 df-uni 4872 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-iota 6464 df-fun 6513 df-fv 6519 df-1st 7968 df-2nd 7969 df-mpst 35480

This theorem is referenced by: msrf 35529 msrid 35532 mthmpps 35569