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Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mpst123 | Structured version Visualization version GIF version |
Description: Decompose a pre-statement into a triple of values. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mpstssv.p | ⊢ 𝑃 = (mPreSt‘𝑇) |
Ref | Expression |
---|---|
mpst123 | ⊢ (𝑋 ∈ 𝑃 → 𝑋 = 〈(1st ‘(1st ‘𝑋)), (2nd ‘(1st ‘𝑋)), (2nd ‘𝑋)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpstssv.p | . . . 4 ⊢ 𝑃 = (mPreSt‘𝑇) | |
2 | 1 | mpstssv 31945 | . . 3 ⊢ 𝑃 ⊆ ((V × V) × V) |
3 | 2 | sseli 3792 | . 2 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ ((V × V) × V)) |
4 | 1st2nd2 7438 | . . . 4 ⊢ (𝑋 ∈ ((V × V) × V) → 𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉) | |
5 | xp1st 7431 | . . . . . 6 ⊢ (𝑋 ∈ ((V × V) × V) → (1st ‘𝑋) ∈ (V × V)) | |
6 | 1st2nd2 7438 | . . . . . 6 ⊢ ((1st ‘𝑋) ∈ (V × V) → (1st ‘𝑋) = 〈(1st ‘(1st ‘𝑋)), (2nd ‘(1st ‘𝑋))〉) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝑋 ∈ ((V × V) × V) → (1st ‘𝑋) = 〈(1st ‘(1st ‘𝑋)), (2nd ‘(1st ‘𝑋))〉) |
8 | 7 | opeq1d 4597 | . . . 4 ⊢ (𝑋 ∈ ((V × V) × V) → 〈(1st ‘𝑋), (2nd ‘𝑋)〉 = 〈〈(1st ‘(1st ‘𝑋)), (2nd ‘(1st ‘𝑋))〉, (2nd ‘𝑋)〉) |
9 | 4, 8 | eqtrd 2831 | . . 3 ⊢ (𝑋 ∈ ((V × V) × V) → 𝑋 = 〈〈(1st ‘(1st ‘𝑋)), (2nd ‘(1st ‘𝑋))〉, (2nd ‘𝑋)〉) |
10 | df-ot 4375 | . . 3 ⊢ 〈(1st ‘(1st ‘𝑋)), (2nd ‘(1st ‘𝑋)), (2nd ‘𝑋)〉 = 〈〈(1st ‘(1st ‘𝑋)), (2nd ‘(1st ‘𝑋))〉, (2nd ‘𝑋)〉 | |
11 | 9, 10 | syl6eqr 2849 | . 2 ⊢ (𝑋 ∈ ((V × V) × V) → 𝑋 = 〈(1st ‘(1st ‘𝑋)), (2nd ‘(1st ‘𝑋)), (2nd ‘𝑋)〉) |
12 | 3, 11 | syl 17 | 1 ⊢ (𝑋 ∈ 𝑃 → 𝑋 = 〈(1st ‘(1st ‘𝑋)), (2nd ‘(1st ‘𝑋)), (2nd ‘𝑋)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 Vcvv 3383 〈cop 4372 〈cotp 4374 × cxp 5308 ‘cfv 6099 1st c1st 7397 2nd c2nd 7398 mPreStcmpst 31879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-sbc 3632 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-ot 4375 df-uni 4627 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-iota 6062 df-fun 6101 df-fv 6107 df-1st 7399 df-2nd 7400 df-mpst 31899 |
This theorem is referenced by: msrf 31948 msrid 31951 mthmpps 31988 |
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