mpst123 - Mathbox for Mario Carneiro

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Theorem mpst123 33498

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 33497	. . 3 ⊢ 𝑃 ⊆ ((V × V) × V)
3	2	sseli 3922	. 2 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ ((V × V) × V))
4		1st2nd2 7863	. . . 4 ⊢ (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
5		xp1st 7856	. . . . . 6 ⊢ (𝑋 ∈ ((V × V) × V) → (1^st ‘𝑋) ∈ (V × V))
6		1st2nd2 7863	. . . . . 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 4816	. . . 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 2780	. . 3 ⊢ (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨⟨(1^st ‘(1^st ‘𝑋)), (2^nd ‘(1^st ‘𝑋))⟩, (2^nd ‘𝑋)⟩)
10		df-ot 4576	. . 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 2798	. 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 1542 ∈ wcel 2110 Vcvv 3431 ⟨cop 4573 ⟨cotp 4575 × cxp 5588 ‘cfv 6432 1^st c1st 7822 2^nd c2nd 7823 mPreStcmpst 33431

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-ot 4576 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-iota 6390 df-fun 6434 df-fv 6440 df-1st 7824 df-2nd 7825 df-mpst 33451

This theorem is referenced by: msrf 33500 msrid 33503 mthmpps 33540