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Mirrors > Home > ILE Home > Th. List > elfvmptrab1 | Unicode version |
Description: Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. Here, the base set of the class abstraction depends on the argument of the function. (Contributed by Alexander van der Vekens, 15-Jul-2018.) |
Ref | Expression |
---|---|
elfvmptrab1.f |
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elfvmptrab1.v |
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Ref | Expression |
---|---|
elfvmptrab1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvmptrab1.f |
. . . . 5
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2 | 1 | funmpt2 5255 |
. . . 4
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3 | funrel 5233 |
. . . 4
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4 | 2, 3 | ax-mp 5 |
. . 3
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5 | relelfvdm 5547 |
. . 3
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6 | 4, 5 | mpan 424 |
. 2
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7 | 1 | dmmptss 5125 |
. . . . . 6
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8 | 7 | sseli 3151 |
. . . . 5
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9 | elfvmptrab1.v |
. . . . . 6
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10 | rabexg 4146 |
. . . . . 6
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11 | 8, 9, 10 | 3syl 17 |
. . . . 5
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12 | nfcv 2319 |
. . . . . 6
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13 | nfsbc1v 2981 |
. . . . . . 7
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14 | nfcv 2319 |
. . . . . . . 8
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15 | 12, 14 | nfcsb 3094 |
. . . . . . 7
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16 | 13, 15 | nfrabxy 2657 |
. . . . . 6
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17 | csbeq1 3060 |
. . . . . . 7
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18 | sbceq1a 2972 |
. . . . . . 7
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19 | 17, 18 | rabeqbidv 2732 |
. . . . . 6
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20 | 12, 16, 19, 1 | fvmptf 5608 |
. . . . 5
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21 | 8, 11, 20 | syl2anc 411 |
. . . 4
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22 | 21 | eleq2d 2247 |
. . 3
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23 | elrabi 2890 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 8, 23 | anim12i 338 |
. . . 4
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25 | 24 | ex 115 |
. . 3
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26 | 22, 25 | sylbid 150 |
. 2
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27 | 6, 26 | mpcom 36 |
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 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fv 5224 |
This theorem is referenced by: elfvmptrab 5611 |
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