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Mirrors > Home > ILE Home > Th. List > tfrlem7 | Unicode version |
Description: Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.) |
Ref | Expression |
---|---|
tfrlem.1 |
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Ref | Expression |
---|---|
tfrlem7 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrlem.1 |
. . 3
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2 | 1 | tfrlem6 6221 |
. 2
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3 | 1 | recsfval 6220 |
. . . . . . . . 9
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4 | 3 | eleq2i 2207 |
. . . . . . . 8
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5 | eluni 3747 |
. . . . . . . 8
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6 | 4, 5 | bitri 183 |
. . . . . . 7
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7 | 3 | eleq2i 2207 |
. . . . . . . 8
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8 | eluni 3747 |
. . . . . . . 8
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9 | 7, 8 | bitri 183 |
. . . . . . 7
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10 | 6, 9 | anbi12i 456 |
. . . . . 6
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11 | eeanv 1905 |
. . . . . 6
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12 | 10, 11 | bitr4i 186 |
. . . . 5
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13 | df-br 3938 |
. . . . . . . . 9
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14 | df-br 3938 |
. . . . . . . . 9
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15 | 13, 14 | anbi12i 456 |
. . . . . . . 8
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16 | 1 | tfrlem5 6219 |
. . . . . . . . 9
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17 | 16 | impcom 124 |
. . . . . . . 8
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18 | 15, 17 | sylanbr 283 |
. . . . . . 7
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19 | 18 | an4s 578 |
. . . . . 6
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20 | 19 | exlimivv 1869 |
. . . . 5
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21 | 12, 20 | sylbi 120 |
. . . 4
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22 | 21 | ax-gen 1426 |
. . 3
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23 | 22 | gen2 1427 |
. 2
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24 | dffun4 5142 |
. 2
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25 | 2, 23, 24 | mpbir2an 927 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-res 4559 df-iota 5096 df-fun 5133 df-fn 5134 df-fv 5139 df-recs 6210 |
This theorem is referenced by: tfrlem9 6224 tfrfun 6225 tfrlemibfn 6233 tfrlemiubacc 6235 tfri1d 6240 rdgfun 6278 |
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