mpst123 - Mathbox for Mario Carneiro

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Theorem mpst123 35903

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 35902	. . 3 ⊢ 𝑃 ⊆ ((V × V) × V)
3	2	sseli 3935	. 2 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ ((V × V) × V))
4		1st2nd2 8013	. . . 4 ⊢ (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
5		xp1st 8006	. . . . . 6 ⊢ (𝑋 ∈ ((V × V) × V) → (1^st ‘𝑋) ∈ (V × V))
6		1st2nd2 8013	. . . . . 6 ⊢ ((1^st ‘𝑋) ∈ (V × V) → (1^st ‘𝑋) = ⟨(1^st ‘(1^st ‘𝑋)), (2^nd ‘(1^st ‘𝑋))⟩)
7	5, 6	syl 18	. . . . 5 ⊢ (𝑋 ∈ ((V × V) × V) → (1^st ‘𝑋) = ⟨(1^st ‘(1^st ‘𝑋)), (2^nd ‘(1^st ‘𝑋))⟩)
8	7	opeq1d 4840	. . . 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 2800	. . 3 ⊢ (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨⟨(1^st ‘(1^st ‘𝑋)), (2^nd ‘(1^st ‘𝑋))⟩, (2^nd ‘𝑋)⟩)
10		df-ot 4594	. . 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 2818	. 2 ⊢ (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨(1^st ‘(1^st ‘𝑋)), (2^nd ‘(1^st ‘𝑋)), (2^nd ‘𝑋)⟩)
12	3, 11	syl 18	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 1563 ∈ wcel 2145 Vcvv 3457 ⟨cop 4591 ⟨cotp 4593 × cxp 5650 ‘cfv 6525 1^st c1st 7972 2^nd c2nd 7973 mPreStcmpst 35836

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3080 df-rex 3090 df-rab 3418 df-v 3459 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-ot 4594 df-uni 4869 df-br 5106 df-opab 5168 df-mpt 5187 df-id 5547 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-iota 6481 df-fun 6527 df-fv 6533 df-1st 7974 df-2nd 7975 df-mpst 35856

This theorem is referenced by: msrf 35905 msrid 35908 mthmpps 35945