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Mirrors > Home > MPE Home > Th. List > Mathboxes > mstapst | Structured version Visualization version GIF version |
Description: A statement is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mstapst.p | β’ π = (mPreStβπ) |
mstapst.s | β’ π = (mStatβπ) |
Ref | Expression |
---|---|
mstapst | β’ π β π |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2730 | . . 3 β’ (mStRedβπ) = (mStRedβπ) | |
2 | mstapst.s | . . 3 β’ π = (mStatβπ) | |
3 | 1, 2 | mstaval 34833 | . 2 β’ π = ran (mStRedβπ) |
4 | mstapst.p | . . . 4 β’ π = (mPreStβπ) | |
5 | 4, 1 | msrf 34831 | . . 3 β’ (mStRedβπ):πβΆπ |
6 | frn 6723 | . . 3 β’ ((mStRedβπ):πβΆπ β ran (mStRedβπ) β π) | |
7 | 5, 6 | ax-mp 5 | . 2 β’ ran (mStRedβπ) β π |
8 | 3, 7 | eqsstri 4015 | 1 β’ π β π |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 β wss 3947 ran crn 5676 βΆwf 6538 βcfv 6542 mPreStcmpst 34762 mStRedcmsr 34763 mStatcmsta 34764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-ot 4636 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 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 7977 df-2nd 7978 df-mpst 34782 df-msr 34783 df-msta 34784 |
This theorem is referenced by: elmsta 34837 mclsssvlem 34851 mclsax 34858 mclsind 34859 mclsppslem 34872 |
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