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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 14121 | . . 3 β’ (π· β (βMetβπ) β π·:(π Γ π)βΆβ*) | |
2 | ffn 5377 | . . 3 β’ (π·:(π Γ π)βΆβ* β π· Fn (π Γ π)) | |
3 | elpreima 5648 | . . 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 4021 | . . 3 β’ (π΄ βΌ π΅ β π΄(β‘π· β β)π΅) |
7 | df-br 4016 | . . 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 981 | . . 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 4668 | . . . . 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 5891 | . . . . 5 β’ (π΄π·π΅) = (π·ββ¨π΄, π΅β©) | |
13 | 12 | eleq1i 2253 | . . . 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 979 = wceq 1363 β wcel 2158 β¨cop 3607 class class class wbr 4015 Γ cxp 4636 β‘ccnv 4637 β cima 4641 Fn wfn 5223 βΆwf 5224 βcfv 5228 (class class class)co 5888 βcr 7823 β*cxr 8004 βMetcxmet 13697 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-map 6663 df-pnf 8007 df-mnf 8008 df-xr 8009 df-xmet 13705 |
This theorem is referenced by: xmeter 14207 xmetec 14208 xmetresbl 14211 |
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