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Mirrors > Home > MPE Home > Th. List > uzin2 | Structured version Visualization version GIF version |
Description: The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.) |
Ref | Expression |
---|---|
uzin2 | β’ ((π΄ β ran β€β₯ β§ π΅ β ran β€β₯) β (π΄ β© π΅) β ran β€β₯) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzf 12829 | . . . 4 β’ β€β₯:β€βΆπ« β€ | |
2 | ffn 6716 | . . . 4 β’ (β€β₯:β€βΆπ« β€ β β€β₯ Fn β€) | |
3 | 1, 2 | ax-mp 5 | . . 3 β’ β€β₯ Fn β€ |
4 | fvelrnb 6951 | . . 3 β’ (β€β₯ Fn β€ β (π΄ β ran β€β₯ β βπ₯ β β€ (β€β₯βπ₯) = π΄)) | |
5 | 3, 4 | ax-mp 5 | . 2 β’ (π΄ β ran β€β₯ β βπ₯ β β€ (β€β₯βπ₯) = π΄) |
6 | fvelrnb 6951 | . . 3 β’ (β€β₯ Fn β€ β (π΅ β ran β€β₯ β βπ¦ β β€ (β€β₯βπ¦) = π΅)) | |
7 | 3, 6 | ax-mp 5 | . 2 β’ (π΅ β ran β€β₯ β βπ¦ β β€ (β€β₯βπ¦) = π΅) |
8 | ineq1 4204 | . . 3 β’ ((β€β₯βπ₯) = π΄ β ((β€β₯βπ₯) β© (β€β₯βπ¦)) = (π΄ β© (β€β₯βπ¦))) | |
9 | 8 | eleq1d 2816 | . 2 β’ ((β€β₯βπ₯) = π΄ β (((β€β₯βπ₯) β© (β€β₯βπ¦)) β ran β€β₯ β (π΄ β© (β€β₯βπ¦)) β ran β€β₯)) |
10 | ineq2 4205 | . . 3 β’ ((β€β₯βπ¦) = π΅ β (π΄ β© (β€β₯βπ¦)) = (π΄ β© π΅)) | |
11 | 10 | eleq1d 2816 | . 2 β’ ((β€β₯βπ¦) = π΅ β ((π΄ β© (β€β₯βπ¦)) β ran β€β₯ β (π΄ β© π΅) β ran β€β₯)) |
12 | uzin 12866 | . . 3 β’ ((π₯ β β€ β§ π¦ β β€) β ((β€β₯βπ₯) β© (β€β₯βπ¦)) = (β€β₯βif(π₯ β€ π¦, π¦, π₯))) | |
13 | ifcl 4572 | . . . . 5 β’ ((π¦ β β€ β§ π₯ β β€) β if(π₯ β€ π¦, π¦, π₯) β β€) | |
14 | 13 | ancoms 457 | . . . 4 β’ ((π₯ β β€ β§ π¦ β β€) β if(π₯ β€ π¦, π¦, π₯) β β€) |
15 | fnfvelrn 7081 | . . . 4 β’ ((β€β₯ Fn β€ β§ if(π₯ β€ π¦, π¦, π₯) β β€) β (β€β₯βif(π₯ β€ π¦, π¦, π₯)) β ran β€β₯) | |
16 | 3, 14, 15 | sylancr 585 | . . 3 β’ ((π₯ β β€ β§ π¦ β β€) β (β€β₯βif(π₯ β€ π¦, π¦, π₯)) β ran β€β₯) |
17 | 12, 16 | eqeltrd 2831 | . 2 β’ ((π₯ β β€ β§ π¦ β β€) β ((β€β₯βπ₯) β© (β€β₯βπ¦)) β ran β€β₯) |
18 | 5, 7, 9, 11, 17 | 2gencl 3515 | 1 β’ ((π΄ β ran β€β₯ β§ π΅ β ran β€β₯) β (π΄ β© π΅) β ran β€β₯) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1539 β wcel 2104 βwrex 3068 β© cin 3946 ifcif 4527 π« cpw 4601 class class class wbr 5147 ran crn 5676 Fn wfn 6537 βΆwf 6538 βcfv 6542 β€ cle 11253 β€cz 12562 β€β₯cuz 12826 |
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-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 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-nel 3045 df-ral 3060 df-rex 3069 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-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 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-ov 7414 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-neg 11451 df-z 12563 df-uz 12827 |
This theorem is referenced by: rexanuz 15296 zfbas 23620 heibor1lem 36980 |
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