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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 31439 | . . . 4 ⊢ 𝑅:𝑃⟶𝑃 |
4 | 3 | ffvelrni 6358 | . . 3 ⊢ (𝑋 ∈ 𝑃 → (𝑅‘𝑋) ∈ 𝑃) |
5 | 4 | a1i 11 | . 2 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑃 → (𝑅‘𝑋) ∈ 𝑃)) |
6 | 3 | ffvelrni 6358 | . . 3 ⊢ (𝑌 ∈ 𝑃 → (𝑅‘𝑌) ∈ 𝑃) |
7 | eleq1 2689 | . . 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 6052 | . . . . . 6 ⊢ dom 𝑅 = 𝑃 |
10 | 0nelxp 5143 | . . . . . . 7 ⊢ ¬ ∅ ∈ ((V × V) × V) | |
11 | 1 | mpstssv 31436 | . . . . . . . 8 ⊢ 𝑃 ⊆ ((V × V) × V) |
12 | 11 | sseli 3599 | . . . . . . 7 ⊢ (∅ ∈ 𝑃 → ∅ ∈ ((V × V) × V)) |
13 | 10, 12 | mto 188 | . . . . . 6 ⊢ ¬ ∅ ∈ 𝑃 |
14 | 9, 13 | ndmfvrcl 6219 | . . . . 5 ⊢ ((𝑅‘𝑋) ∈ 𝑃 → 𝑋 ∈ 𝑃) |
15 | 14 | adantl 482 | . . . 4 ⊢ (((𝑅‘𝑋) = (𝑅‘𝑌) ∧ (𝑅‘𝑋) ∈ 𝑃) → 𝑋 ∈ 𝑃) |
16 | 7 | biimpa 501 | . . . . 5 ⊢ (((𝑅‘𝑋) = (𝑅‘𝑌) ∧ (𝑅‘𝑋) ∈ 𝑃) → (𝑅‘𝑌) ∈ 𝑃) |
17 | 9, 13 | ndmfvrcl 6219 | . . . . 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 450 | . 2 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → ((𝑅‘𝑋) ∈ 𝑃 → (𝑋 ∈ 𝑃 ↔ 𝑌 ∈ 𝑃))) |
21 | 5, 8, 20 | pm5.21ndd 369 | 1 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑃 ↔ 𝑌 ∈ 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 × cxp 5112 ‘cfv 5888 mPreStcmpst 31370 mStRedcmsr 31371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-ot 4186 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-1st 7168 df-2nd 7169 df-mpst 31390 df-msr 31391 |
This theorem is referenced by: elmthm 31473 |
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