mpst123 - Mathbox for Mario Carneiro

	Mathbox for Mario Carneiro		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > mpst123		Structured version Visualization version GIF version

Theorem mpst123 36472

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 36471	. . 3 ⊢ 𝑃 ⊆ ((V × V) × V)
3	2	sseli 3909	. 2 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ ((V × V) × V))
4		1st2nd2 8064	. . . 4 ⊢ (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
5		xp1st 8057	. . . . . 6 ⊢ (𝑋 ∈ ((V × V) × V) → (1^st ‘𝑋) ∈ (V × V))
6		1st2nd2 8064	. . . . . 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 4849	. . . 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 4862	. . 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

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2147 Vcvv 3454 ⟨cop 4838 ⟨cotp 4861 × cxp 5653 ‘cfv 6558 1^st c1st 8023 2^nd c2nd 8024 mPreStcmpst 36405

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 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775

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-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-ot 4862 df-uni 4871 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-iota 6512 df-fun 6560 df-fv 6566 df-1st 8025 df-2nd 8026 df-mpst 36425

This theorem is used by: msrf 36474 msrid 36477 mthmpps 36514

W3C validator