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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 32899 | . . 3 ⊢ 𝑃 ⊆ ((V × V) × V) |
3 | 2 | sseli 3911 | . 2 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ ((V × V) × V)) |
4 | 1st2nd2 7710 | . . . 4 ⊢ (𝑋 ∈ ((V × V) × V) → 𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉) | |
5 | xp1st 7703 | . . . . . 6 ⊢ (𝑋 ∈ ((V × V) × V) → (1st ‘𝑋) ∈ (V × V)) | |
6 | 1st2nd2 7710 | . . . . . 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 4771 | . . . 4 ⊢ (𝑋 ∈ ((V × V) × V) → 〈(1st ‘𝑋), (2nd ‘𝑋)〉 = 〈〈(1st ‘(1st ‘𝑋)), (2nd ‘(1st ‘𝑋))〉, (2nd ‘𝑋)〉) |
9 | 4, 8 | eqtrd 2833 | . . 3 ⊢ (𝑋 ∈ ((V × V) × V) → 𝑋 = 〈〈(1st ‘(1st ‘𝑋)), (2nd ‘(1st ‘𝑋))〉, (2nd ‘𝑋)〉) |
10 | df-ot 4534 | . . 3 ⊢ 〈(1st ‘(1st ‘𝑋)), (2nd ‘(1st ‘𝑋)), (2nd ‘𝑋)〉 = 〈〈(1st ‘(1st ‘𝑋)), (2nd ‘(1st ‘𝑋))〉, (2nd ‘𝑋)〉 | |
11 | 9, 10 | eqtr4di 2851 | . 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 1538 ∈ wcel 2111 Vcvv 3441 〈cop 4531 〈cotp 4533 × cxp 5517 ‘cfv 6324 1st c1st 7669 2nd c2nd 7670 mPreStcmpst 32833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-ot 4534 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fv 6332 df-1st 7671 df-2nd 7672 df-mpst 32853 |
This theorem is referenced by: msrf 32902 msrid 32905 mthmpps 32942 |
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