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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 32786 | . . 3 ⊢ 𝑃 ⊆ ((V × V) × V) |
3 | 2 | sseli 3963 | . 2 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ ((V × V) × V)) |
4 | 1st2nd2 7728 | . . . 4 ⊢ (𝑋 ∈ ((V × V) × V) → 𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉) | |
5 | xp1st 7721 | . . . . . 6 ⊢ (𝑋 ∈ ((V × V) × V) → (1st ‘𝑋) ∈ (V × V)) | |
6 | 1st2nd2 7728 | . . . . . 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 4809 | . . . 4 ⊢ (𝑋 ∈ ((V × V) × V) → 〈(1st ‘𝑋), (2nd ‘𝑋)〉 = 〈〈(1st ‘(1st ‘𝑋)), (2nd ‘(1st ‘𝑋))〉, (2nd ‘𝑋)〉) |
9 | 4, 8 | eqtrd 2856 | . . 3 ⊢ (𝑋 ∈ ((V × V) × V) → 𝑋 = 〈〈(1st ‘(1st ‘𝑋)), (2nd ‘(1st ‘𝑋))〉, (2nd ‘𝑋)〉) |
10 | df-ot 4576 | . . 3 ⊢ 〈(1st ‘(1st ‘𝑋)), (2nd ‘(1st ‘𝑋)), (2nd ‘𝑋)〉 = 〈〈(1st ‘(1st ‘𝑋)), (2nd ‘(1st ‘𝑋))〉, (2nd ‘𝑋)〉 | |
11 | 9, 10 | syl6eqr 2874 | . 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 1537 ∈ wcel 2114 Vcvv 3494 〈cop 4573 〈cotp 4575 × cxp 5553 ‘cfv 6355 1st c1st 7687 2nd c2nd 7688 mPreStcmpst 32720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-ot 4576 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fv 6363 df-1st 7689 df-2nd 7690 df-mpst 32740 |
This theorem is referenced by: msrf 32789 msrid 32792 mthmpps 32829 |
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