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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 35146 | . . . 4 ⊢ 𝑅:𝑃⟶𝑃 |
4 | 3 | ffvelcdmi 7087 | . . 3 ⊢ (𝑋 ∈ 𝑃 → (𝑅‘𝑋) ∈ 𝑃) |
5 | 4 | a1i 11 | . 2 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑃 → (𝑅‘𝑋) ∈ 𝑃)) |
6 | 3 | ffvelcdmi 7087 | . . 3 ⊢ (𝑌 ∈ 𝑃 → (𝑅‘𝑌) ∈ 𝑃) |
7 | eleq1 2817 | . . 3 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → ((𝑅‘𝑋) ∈ 𝑃 ↔ (𝑅‘𝑌) ∈ 𝑃)) | |
8 | 6, 7 | imbitrrid 245 | . 2 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑌 ∈ 𝑃 → (𝑅‘𝑋) ∈ 𝑃)) |
9 | 3 | fdmi 6728 | . . . . . 6 ⊢ dom 𝑅 = 𝑃 |
10 | 0nelxp 5706 | . . . . . . 7 ⊢ ¬ ∅ ∈ ((V × V) × V) | |
11 | 1 | mpstssv 35143 | . . . . . . . 8 ⊢ 𝑃 ⊆ ((V × V) × V) |
12 | 11 | sseli 3974 | . . . . . . 7 ⊢ (∅ ∈ 𝑃 → ∅ ∈ ((V × V) × V)) |
13 | 10, 12 | mto 196 | . . . . . 6 ⊢ ¬ ∅ ∈ 𝑃 |
14 | 9, 13 | ndmfvrcl 6927 | . . . . 5 ⊢ ((𝑅‘𝑋) ∈ 𝑃 → 𝑋 ∈ 𝑃) |
15 | 14 | adantl 481 | . . . 4 ⊢ (((𝑅‘𝑋) = (𝑅‘𝑌) ∧ (𝑅‘𝑋) ∈ 𝑃) → 𝑋 ∈ 𝑃) |
16 | 7 | biimpa 476 | . . . . 5 ⊢ (((𝑅‘𝑋) = (𝑅‘𝑌) ∧ (𝑅‘𝑋) ∈ 𝑃) → (𝑅‘𝑌) ∈ 𝑃) |
17 | 9, 13 | ndmfvrcl 6927 | . . . . 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 265 | . . 3 ⊢ (((𝑅‘𝑋) = (𝑅‘𝑌) ∧ (𝑅‘𝑋) ∈ 𝑃) → (𝑋 ∈ 𝑃 ↔ 𝑌 ∈ 𝑃)) |
20 | 19 | ex 412 | . 2 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → ((𝑅‘𝑋) ∈ 𝑃 → (𝑋 ∈ 𝑃 ↔ 𝑌 ∈ 𝑃))) |
21 | 5, 8, 20 | pm5.21ndd 379 | 1 ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑃 ↔ 𝑌 ∈ 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3470 ∅c0 4318 × cxp 5670 ‘cfv 6542 mPreStcmpst 35077 mStRedcmsr 35078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-ot 4633 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-1st 7987 df-2nd 7988 df-mpst 35097 df-msr 35098 |
This theorem is referenced by: elmthm 35180 |
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