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Mirrors > Home > MPE Home > Th. List > elfg | Structured version Visualization version Unicode version |
Description: A condition for elements of a generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
Ref | Expression |
---|---|
elfg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fgval 20878 |
. . 3
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2 | 1 | eleq2d 2513 |
. 2
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3 | pweq 3953 |
. . . . . 6
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4 | 3 | ineq2d 3633 |
. . . . 5
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5 | 4 | neeq1d 2682 |
. . . 4
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6 | 5 | elrab 3195 |
. . 3
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7 | elfvdm 5889 |
. . . . 5
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8 | elpw2g 4565 |
. . . . 5
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9 | 7, 8 | syl 17 |
. . . 4
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10 | elin 3616 |
. . . . . . . 8
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11 | selpw 3957 |
. . . . . . . . 9
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12 | 11 | anbi2i 699 |
. . . . . . . 8
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13 | 10, 12 | bitri 253 |
. . . . . . 7
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14 | 13 | exbii 1717 |
. . . . . 6
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15 | n0 3740 |
. . . . . 6
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16 | df-rex 2742 |
. . . . . 6
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17 | 14, 15, 16 | 3bitr4i 281 |
. . . . 5
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18 | 17 | a1i 11 |
. . . 4
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19 | 9, 18 | anbi12d 716 |
. . 3
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20 | 6, 19 | syl5bb 261 |
. 2
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21 | 2, 20 | bitrd 257 |
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 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-ral 2741 df-rex 2742 df-rab 2745 df-v 3046 df-sbc 3267 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-op 3974 df-uni 4198 df-br 4402 df-opab 4461 df-id 4748 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-iota 5545 df-fun 5583 df-fv 5589 df-ov 6291 df-oprab 6292 df-mpt2 6293 df-fg 18961 |
This theorem is referenced by: ssfg 20880 fgss 20881 fgss2 20882 fgfil 20883 elfilss 20884 fgcl 20886 fgabs 20887 fgtr 20898 trfg 20899 uffix 20929 elfm 20955 elfm2 20956 elfm3 20958 fbflim 20984 flffbas 21003 fclsbas 21029 isucn2 21287 metust 21566 cfilucfil 21567 metuel 21572 fgcfil 22234 fgmin 31019 filnetlem4 31030 |
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