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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 12773 | . . . 4 β’ β€β₯:β€βΆπ« β€ | |
2 | ffn 6673 | . . . 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 6908 | . . 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 6908 | . . 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 4170 | . . 3 β’ ((β€β₯βπ₯) = π΄ β ((β€β₯βπ₯) β© (β€β₯βπ¦)) = (π΄ β© (β€β₯βπ¦))) | |
9 | 8 | eleq1d 2823 | . 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 4171 | . . 3 β’ ((β€β₯βπ¦) = π΅ β (π΄ β© (β€β₯βπ¦)) = (π΄ β© π΅)) | |
11 | 10 | eleq1d 2823 | . 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 12810 | . . 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 4536 | . . . . 5 β’ ((π¦ β β€ β§ π₯ β β€) β if(π₯ β€ π¦, π¦, π₯) β β€) | |
14 | 13 | ancoms 460 | . . . 4 β’ ((π₯ β β€ β§ π¦ β β€) β if(π₯ β€ π¦, π¦, π₯) β β€) |
15 | fnfvelrn 7036 | . . . 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 588 | . . 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 2838 | . 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 3489 | 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 397 = wceq 1542 β wcel 2107 βwrex 3074 β© cin 3914 ifcif 4491 π« cpw 4565 class class class wbr 5110 ran crn 5639 Fn wfn 6496 βΆwf 6497 βcfv 6501 β€ cle 11197 β€cz 12506 β€β₯cuz 12770 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-pre-lttri 11132 ax-pre-lttrn 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-po 5550 df-so 5551 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-neg 11395 df-z 12507 df-uz 12771 |
This theorem is referenced by: rexanuz 15237 zfbas 23263 heibor1lem 36297 |
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