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Mirrors > Home > ILE Home > Th. List > xmeterval | GIF version |
Description: Value of the "finitely separated" relation. (Contributed by Mario Carneiro, 24-Aug-2015.) |
Ref | Expression |
---|---|
xmeter.1 | β’ βΌ = (β‘π· β β) |
Ref | Expression |
---|---|
xmeterval | β’ (π· β (βMetβπ) β (π΄ βΌ π΅ β (π΄ β π β§ π΅ β π β§ (π΄π·π΅) β β))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xmetf 14253 | . . 3 β’ (π· β (βMetβπ) β π·:(π Γ π)βΆβ*) | |
2 | ffn 5380 | . . 3 β’ (π·:(π Γ π)βΆβ* β π· Fn (π Γ π)) | |
3 | elpreima 5651 | . . 3 β’ (π· Fn (π Γ π) β (β¨π΄, π΅β© β (β‘π· β β) β (β¨π΄, π΅β© β (π Γ π) β§ (π·ββ¨π΄, π΅β©) β β))) | |
4 | 1, 2, 3 | 3syl 17 | . 2 β’ (π· β (βMetβπ) β (β¨π΄, π΅β© β (β‘π· β β) β (β¨π΄, π΅β© β (π Γ π) β§ (π·ββ¨π΄, π΅β©) β β))) |
5 | xmeter.1 | . . . 4 β’ βΌ = (β‘π· β β) | |
6 | 5 | breqi 4024 | . . 3 β’ (π΄ βΌ π΅ β π΄(β‘π· β β)π΅) |
7 | df-br 4019 | . . 3 β’ (π΄(β‘π· β β)π΅ β β¨π΄, π΅β© β (β‘π· β β)) | |
8 | 6, 7 | bitri 184 | . 2 β’ (π΄ βΌ π΅ β β¨π΄, π΅β© β (β‘π· β β)) |
9 | df-3an 982 | . . 3 β’ ((π΄ β π β§ π΅ β π β§ (π΄π·π΅) β β) β ((π΄ β π β§ π΅ β π) β§ (π΄π·π΅) β β)) | |
10 | opelxp 4671 | . . . . 5 β’ (β¨π΄, π΅β© β (π Γ π) β (π΄ β π β§ π΅ β π)) | |
11 | 10 | bicomi 132 | . . . 4 β’ ((π΄ β π β§ π΅ β π) β β¨π΄, π΅β© β (π Γ π)) |
12 | df-ov 5894 | . . . . 5 β’ (π΄π·π΅) = (π·ββ¨π΄, π΅β©) | |
13 | 12 | eleq1i 2255 | . . . 4 β’ ((π΄π·π΅) β β β (π·ββ¨π΄, π΅β©) β β) |
14 | 11, 13 | anbi12i 460 | . . 3 β’ (((π΄ β π β§ π΅ β π) β§ (π΄π·π΅) β β) β (β¨π΄, π΅β© β (π Γ π) β§ (π·ββ¨π΄, π΅β©) β β)) |
15 | 9, 14 | bitri 184 | . 2 β’ ((π΄ β π β§ π΅ β π β§ (π΄π·π΅) β β) β (β¨π΄, π΅β© β (π Γ π) β§ (π·ββ¨π΄, π΅β©) β β)) |
16 | 4, 8, 15 | 3bitr4g 223 | 1 β’ (π· β (βMetβπ) β (π΄ βΌ π΅ β (π΄ β π β§ π΅ β π β§ (π΄π·π΅) β β))) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 β wb 105 β§ w3a 980 = wceq 1364 β wcel 2160 β¨cop 3610 class class class wbr 4018 Γ cxp 4639 β‘ccnv 4640 β cima 4644 Fn wfn 5226 βΆwf 5227 βcfv 5231 (class class class)co 5891 βcr 7829 β*cxr 8010 βMetcxmet 13816 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7921 ax-resscn 7922 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-map 6668 df-pnf 8013 df-mnf 8014 df-xr 8015 df-xmet 13824 |
This theorem is referenced by: xmeter 14339 xmetec 14340 xmetresbl 14343 |
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