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Mirrors > Home > MPE Home > Th. List > Mathboxes > msrrcl | Structured version Visualization version GIF version |
Description: If 𝑋 and 𝑌 have the same reduct, then one is a pre-statement iff the other is. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mpstssv.p | ⊢ 𝑃 = (mPreSt‘𝑇) |
msrf.r | ⊢ 𝑅 = (mStRed‘𝑇) |
Ref | Expression |
---|---|
msrrcl | ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑃 ↔ 𝑌 ∈ 𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpstssv.p | . . . . 5 ⊢ 𝑃 = (mPreSt‘𝑇) | |
2 | msrf.r | . . . . 5 ⊢ 𝑅 = (mStRed‘𝑇) | |
3 | 1, 2 | msrf 31746 | . . . 4 ⊢ 𝑅:𝑃⟶𝑃 |
4 | 3 | ffvelrni 6521 | . . 3 ⊢ (𝑋 ∈ 𝑃 → (𝑅‘𝑋) ∈ 𝑃) |
5 | 4 | a1i 11 | . 2 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑃 → (𝑅‘𝑋) ∈ 𝑃)) |
6 | 3 | ffvelrni 6521 | . . 3 ⊢ (𝑌 ∈ 𝑃 → (𝑅‘𝑌) ∈ 𝑃) |
7 | eleq1 2827 | . . 3 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → ((𝑅‘𝑋) ∈ 𝑃 ↔ (𝑅‘𝑌) ∈ 𝑃)) | |
8 | 6, 7 | syl5ibr 236 | . 2 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑌 ∈ 𝑃 → (𝑅‘𝑋) ∈ 𝑃)) |
9 | 3 | fdmi 6213 | . . . . . 6 ⊢ dom 𝑅 = 𝑃 |
10 | 0nelxp 5300 | . . . . . . 7 ⊢ ¬ ∅ ∈ ((V × V) × V) | |
11 | 1 | mpstssv 31743 | . . . . . . . 8 ⊢ 𝑃 ⊆ ((V × V) × V) |
12 | 11 | sseli 3740 | . . . . . . 7 ⊢ (∅ ∈ 𝑃 → ∅ ∈ ((V × V) × V)) |
13 | 10, 12 | mto 188 | . . . . . 6 ⊢ ¬ ∅ ∈ 𝑃 |
14 | 9, 13 | ndmfvrcl 6380 | . . . . 5 ⊢ ((𝑅‘𝑋) ∈ 𝑃 → 𝑋 ∈ 𝑃) |
15 | 14 | adantl 473 | . . . 4 ⊢ (((𝑅‘𝑋) = (𝑅‘𝑌) ∧ (𝑅‘𝑋) ∈ 𝑃) → 𝑋 ∈ 𝑃) |
16 | 7 | biimpa 502 | . . . . 5 ⊢ (((𝑅‘𝑋) = (𝑅‘𝑌) ∧ (𝑅‘𝑋) ∈ 𝑃) → (𝑅‘𝑌) ∈ 𝑃) |
17 | 9, 13 | ndmfvrcl 6380 | . . . . 5 ⊢ ((𝑅‘𝑌) ∈ 𝑃 → 𝑌 ∈ 𝑃) |
18 | 16, 17 | syl 17 | . . . 4 ⊢ (((𝑅‘𝑋) = (𝑅‘𝑌) ∧ (𝑅‘𝑋) ∈ 𝑃) → 𝑌 ∈ 𝑃) |
19 | 15, 18 | 2thd 255 | . . 3 ⊢ (((𝑅‘𝑋) = (𝑅‘𝑌) ∧ (𝑅‘𝑋) ∈ 𝑃) → (𝑋 ∈ 𝑃 ↔ 𝑌 ∈ 𝑃)) |
20 | 19 | ex 449 | . 2 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → ((𝑅‘𝑋) ∈ 𝑃 → (𝑋 ∈ 𝑃 ↔ 𝑌 ∈ 𝑃))) |
21 | 5, 8, 20 | pm5.21ndd 368 | 1 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑃 ↔ 𝑌 ∈ 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ∅c0 4058 × cxp 5264 ‘cfv 6049 mPreStcmpst 31677 mStRedcmsr 31678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-ot 4330 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-1st 7333 df-2nd 7334 df-mpst 31697 df-msr 31698 |
This theorem is referenced by: elmthm 31780 |
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