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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 3036 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | eqcom 2102 |
. . . . . . . . . 10
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3 | ssel 3041 |
. . . . . . . . . . . 12
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4 | funbrfvb 5396 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 4 | ex 114 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | 3, 5 | syl9 72 |
. . . . . . . . . . 11
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7 | 6 | imp31 254 |
. . . . . . . . . 10
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8 | 2, 7 | syl5bb 191 |
. . . . . . . . 9
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9 | 8 | rexbidva 2393 |
. . . . . . . 8
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10 | vex 2644 |
. . . . . . . . 9
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11 | 10 | elima 4822 |
. . . . . . . 8
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12 | 9, 11 | syl6rbbr 198 |
. . . . . . 7
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13 | 12 | imbi1d 230 |
. . . . . 6
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14 | r19.23v 2500 |
. . . . . 6
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15 | 13, 14 | syl6bbr 197 |
. . . . 5
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16 | 15 | albidv 1763 |
. . . 4
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17 | 16 | ancoms 266 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | ralcom4 2663 |
. . . 4
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19 | ssel2 3042 |
. . . . . . . . 9
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20 | 19 | anim2i 337 |
. . . . . . . 8
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21 | 20 | 3impb 1145 |
. . . . . . 7
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22 | funfvex 5370 |
. . . . . . 7
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23 | nfv 1476 |
. . . . . . . 8
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24 | eleq1 2162 |
. . . . . . . 8
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25 | 23, 24 | ceqsalg 2669 |
. . . . . . 7
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26 | 21, 22, 25 | 3syl 17 |
. . . . . 6
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27 | 26 | 3expa 1149 |
. . . . 5
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28 | 27 | ralbidva 2392 |
. . . 4
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29 | 18, 28 | syl5bbr 193 |
. . 3
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30 | 17, 29 | bitrd 187 |
. 2
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31 | 1, 30 | syl5bb 191 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-sbc 2863 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-fv 5067 |
This theorem is referenced by: funimass3 5468 funimass5 5469 funconstss 5470 funimassov 5852 phimullem 11693 txcnp 12221 metcnp 12436 |
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