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Mirrors > Home > MPE Home > Th. List > Mathboxes > msrrcl | Structured version Visualization version GIF version |
Description: If π and π 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 | β’ π = (mPreStβπ) |
msrf.r | β’ π = (mStRedβπ) |
Ref | Expression |
---|---|
msrrcl | β’ ((π βπ) = (π βπ) β (π β π β π β π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpstssv.p | . . . . 5 β’ π = (mPreStβπ) | |
2 | msrf.r | . . . . 5 β’ π = (mStRedβπ) | |
3 | 1, 2 | msrf 34200 | . . . 4 β’ π :πβΆπ |
4 | 3 | ffvelcdmi 7038 | . . 3 β’ (π β π β (π βπ) β π) |
5 | 4 | a1i 11 | . 2 β’ ((π βπ) = (π βπ) β (π β π β (π βπ) β π)) |
6 | 3 | ffvelcdmi 7038 | . . 3 β’ (π β π β (π βπ) β π) |
7 | eleq1 2822 | . . 3 β’ ((π βπ) = (π βπ) β ((π βπ) β π β (π βπ) β π)) | |
8 | 6, 7 | syl5ibr 246 | . 2 β’ ((π βπ) = (π βπ) β (π β π β (π βπ) β π)) |
9 | 3 | fdmi 6684 | . . . . . 6 β’ dom π = π |
10 | 0nelxp 5671 | . . . . . . 7 β’ Β¬ β β ((V Γ V) Γ V) | |
11 | 1 | mpstssv 34197 | . . . . . . . 8 β’ π β ((V Γ V) Γ V) |
12 | 11 | sseli 3944 | . . . . . . 7 β’ (β β π β β β ((V Γ V) Γ V)) |
13 | 10, 12 | mto 196 | . . . . . 6 β’ Β¬ β β π |
14 | 9, 13 | ndmfvrcl 6882 | . . . . 5 β’ ((π βπ) β π β π β π) |
15 | 14 | adantl 483 | . . . 4 β’ (((π βπ) = (π βπ) β§ (π βπ) β π) β π β π) |
16 | 7 | biimpa 478 | . . . . 5 β’ (((π βπ) = (π βπ) β§ (π βπ) β π) β (π βπ) β π) |
17 | 9, 13 | ndmfvrcl 6882 | . . . . 5 β’ ((π βπ) β π β π β π) |
18 | 16, 17 | syl 17 | . . . 4 β’ (((π βπ) = (π βπ) β§ (π βπ) β π) β π β π) |
19 | 15, 18 | 2thd 265 | . . 3 β’ (((π βπ) = (π βπ) β§ (π βπ) β π) β (π β π β π β π)) |
20 | 19 | ex 414 | . 2 β’ ((π βπ) = (π βπ) β ((π βπ) β π β (π β π β π β π))) |
21 | 5, 8, 20 | pm5.21ndd 381 | 1 β’ ((π βπ) = (π βπ) β (π β π β π β π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3447 β c0 4286 Γ cxp 5635 βcfv 6500 mPreStcmpst 34131 mStRedcmsr 34132 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-ot 4599 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-1st 7925 df-2nd 7926 df-mpst 34151 df-msr 34152 |
This theorem is referenced by: elmthm 34234 |
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