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Mirrors > Home > MPE Home > Th. List > elfvmptrab1 | Structured version Visualization version 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 | ne0i 3728 |
. . 3
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2 | ndmfv 5903 |
. . . 4
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3 | 2 | necon1ai 2670 |
. . 3
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4 | elfvmptrab1.f |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 4 | dmmptss 5338 |
. . . . . . 7
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6 | 5 | sseli 3414 |
. . . . . 6
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7 | elfvmptrab1.v |
. . . . . . 7
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8 | rabexg 4549 |
. . . . . . 7
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9 | 6, 7, 8 | 3syl 18 |
. . . . . 6
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10 | nfcv 2612 |
. . . . . . 7
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11 | nfsbc1v 3275 |
. . . . . . . 8
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12 | nfcv 2612 |
. . . . . . . . 9
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13 | 10, 12 | nfcsb 3367 |
. . . . . . . 8
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14 | 11, 13 | nfrab 2958 |
. . . . . . 7
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15 | csbeq1 3352 |
. . . . . . . 8
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16 | sbceq1a 3266 |
. . . . . . . 8
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17 | 15, 16 | rabeqbidv 3026 |
. . . . . . 7
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18 | 10, 14, 17, 4 | fvmptf 5981 |
. . . . . 6
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19 | 6, 9, 18 | syl2anc 673 |
. . . . 5
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20 | 19 | eleq2d 2534 |
. . . 4
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21 | elrabi 3181 |
. . . . . 6
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22 | 6, 21 | anim12i 576 |
. . . . 5
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23 | 22 | ex 441 |
. . . 4
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24 | 20, 23 | sylbid 223 |
. . 3
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25 | 1, 3, 24 | 3syl 18 |
. 2
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26 | 25 | pm2.43i 48 |
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 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-ral 2761 df-rex 2762 df-rab 2765 df-v 3033 df-sbc 3256 df-csb 3350 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-sn 3960 df-pr 3962 df-op 3966 df-uni 4191 df-br 4396 df-opab 4455 df-mpt 4456 df-id 4754 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 df-iota 5553 df-fun 5591 df-fv 5597 |
This theorem is referenced by: elfvmptrab 5986 |
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