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Mirrors > Home > ILE Home > Th. List > funimass4 | Unicode version |
Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Raph Levien, 20-Nov-2006.) |
Ref | Expression |
---|---|
funimass4 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 2928 |
. 2
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2 | eqcom 2039 |
. . . . . . . . . 10
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3 | ssel 2933 |
. . . . . . . . . . . 12
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4 | funbrfvb 5159 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 4 | ex 108 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | 3, 5 | syl9 66 |
. . . . . . . . . . 11
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7 | 6 | imp31 243 |
. . . . . . . . . 10
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8 | 2, 7 | syl5bb 181 |
. . . . . . . . 9
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9 | 8 | rexbidva 2317 |
. . . . . . . 8
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10 | vex 2554 |
. . . . . . . . 9
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11 | 10 | elima 4616 |
. . . . . . . 8
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12 | 9, 11 | syl6rbbr 188 |
. . . . . . 7
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13 | 12 | imbi1d 220 |
. . . . . 6
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14 | r19.23v 2419 |
. . . . . 6
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15 | 13, 14 | syl6bbr 187 |
. . . . 5
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16 | 15 | albidv 1702 |
. . . 4
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17 | 16 | ancoms 255 |
. . 3
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18 | ralcom4 2570 |
. . . 4
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19 | ssel2 2934 |
. . . . . . . . 9
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20 | 19 | anim2i 324 |
. . . . . . . 8
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21 | 20 | 3impb 1099 |
. . . . . . 7
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22 | funfvex 5135 |
. . . . . . 7
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23 | nfv 1418 |
. . . . . . . 8
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24 | eleq1 2097 |
. . . . . . . 8
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25 | 23, 24 | ceqsalg 2576 |
. . . . . . 7
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26 | 21, 22, 25 | 3syl 17 |
. . . . . 6
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27 | 26 | 3expa 1103 |
. . . . 5
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28 | 27 | ralbidva 2316 |
. . . 4
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29 | 18, 28 | syl5bbr 183 |
. . 3
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30 | 17, 29 | bitrd 177 |
. 2
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31 | 1, 30 | syl5bb 181 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-fv 4853 |
This theorem is referenced by: funimass3 5226 funimass5 5227 funconstss 5228 funimassov 5592 |
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