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Mirrors > Home > MPE Home > Th. List > cfilufg | Structured version Unicode version |
Description: The filter generated by a Cauchy filter base is still a Cauchy filter base. (Contributed by Thierry Arnoux, 24-Jan-2018.) |
Ref | Expression |
---|---|
cfilufg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cfilufbas 19980 |
. . 3
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2 | fgcl 19567 |
. . 3
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3 | filfbas 19537 |
. . 3
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4 | 1, 2, 3 | 3syl 20 |
. 2
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5 | 1 | ad3antrrr 729 |
. . . . . . 7
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6 | ssfg 19561 |
. . . . . . 7
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7 | 5, 6 | syl 16 |
. . . . . 6
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8 | simplr 754 |
. . . . . 6
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9 | 7, 8 | sseldd 3455 |
. . . . 5
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10 | id 22 |
. . . . . . . 8
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11 | 10, 10 | xpeq12d 4963 |
. . . . . . 7
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12 | 11 | sseq1d 3481 |
. . . . . 6
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13 | 12 | rspcev 3169 |
. . . . 5
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14 | 9, 13 | sylancom 667 |
. . . 4
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15 | iscfilu 19979 |
. . . . . 6
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16 | 15 | simplbda 624 |
. . . . 5
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17 | 16 | r19.21bi 2910 |
. . . 4
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18 | 14, 17 | r19.29a 2958 |
. . 3
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19 | 18 | ralrimiva 2822 |
. 2
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20 | iscfilu 19979 |
. . 3
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21 | 20 | adantr 465 |
. 2
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22 | 4, 19, 21 | mpbir2and 913 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-sep 4511 ax-nul 4519 ax-pow 4568 ax-pr 4629 ax-un 6472 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-nel 2647 df-ral 2800 df-rex 2801 df-rab 2804 df-v 3070 df-sbc 3285 df-csb 3387 df-dif 3429 df-un 3431 df-in 3433 df-ss 3440 df-nul 3736 df-if 3890 df-pw 3960 df-sn 3976 df-pr 3978 df-op 3982 df-uni 4190 df-br 4391 df-opab 4449 df-mpt 4450 df-id 4734 df-xp 4944 df-rel 4945 df-cnv 4946 df-co 4947 df-dm 4948 df-rn 4949 df-res 4950 df-ima 4951 df-iota 5479 df-fun 5518 df-fn 5519 df-fv 5524 df-ov 6193 df-oprab 6194 df-mpt2 6195 df-fbas 17923 df-fg 17924 df-fil 19535 df-ust 19891 df-cfilu 19978 |
This theorem is referenced by: ucnextcn 19995 |
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