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Mirrors > Home > MPE Home > Th. List > isf34lem2 | Structured version Visualization version GIF version |
Description: Lemma for isfin3-4 10405. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
Ref | Expression |
---|---|
compss.a | β’ πΉ = (π₯ β π« π΄ β¦ (π΄ β π₯)) |
Ref | Expression |
---|---|
isf34lem2 | β’ (π΄ β π β πΉ:π« π΄βΆπ« π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 4124 | . . . 4 β’ (π΄ β π₯) β π΄ | |
2 | elpw2g 5341 | . . . 4 β’ (π΄ β π β ((π΄ β π₯) β π« π΄ β (π΄ β π₯) β π΄)) | |
3 | 1, 2 | mpbiri 257 | . . 3 β’ (π΄ β π β (π΄ β π₯) β π« π΄) |
4 | 3 | adantr 479 | . 2 β’ ((π΄ β π β§ π₯ β π« π΄) β (π΄ β π₯) β π« π΄) |
5 | compss.a | . 2 β’ πΉ = (π₯ β π« π΄ β¦ (π΄ β π₯)) | |
6 | 4, 5 | fmptd 7119 | 1 β’ (π΄ β π β πΉ:π« π΄βΆπ« π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β cdif 3936 β wss 3939 π« cpw 4598 β¦ cmpt 5226 βΆwf 6539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-br 5144 df-opab 5206 df-mpt 5227 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-fun 6545 df-fn 6546 df-f 6547 |
This theorem is referenced by: isf34lem5 10401 isf34lem7 10402 isf34lem6 10403 |
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