mpst123 - Mathbox for Mario Carneiro

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Theorem mpst123 36010

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 36009	. . 3 ⊢ 𝑃 ⊆ ((V × V) × V)
3	2	sseli 3934	. 2 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ ((V × V) × V))
4		1st2nd2 8026	. . . 4 ⊢ (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
5		xp1st 8019	. . . . . 6 ⊢ (𝑋 ∈ ((V × V) × V) → (1^st ‘𝑋) ∈ (V × V))
6		1st2nd2 8026	. . . . . 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 4845	. . . 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 2798	. . 3 ⊢ (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨⟨(1^st ‘(1^st ‘𝑋)), (2^nd ‘(1^st ‘𝑋))⟩, (2^nd ‘𝑋)⟩)
10		df-ot 4599	. . 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 2816	. 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 1570 ∈ wcel 2143 Vcvv 3455 ⟨cop 4596 ⟨cotp 4598 × cxp 5661 ‘cfv 6538 1^st c1st 7985 2^nd c2nd 7986 mPreStcmpst 35943

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-ot 4599 df-uni 4874 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-iota 6494 df-fun 6540 df-fv 6546 df-1st 7987 df-2nd 7988 df-mpst 35963

This theorem is referenced by: msrf 36012 msrid 36015 mthmpps 36052