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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > afv0nbfvbi | Structured version Visualization version Unicode version |
Description: The function's value at an argument is an element of a set if and only if the value of the alternative function at this argument is an element of that set, if the set does not contain the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.) |
Ref | Expression |
---|---|
afv0nbfvbi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | afvvfveq 38688 |
. . 3
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2 | eleq1 2528 |
. . . 4
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3 | 2 | biimpd 212 |
. . 3
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4 | 1, 3 | mpcom 37 |
. 2
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5 | elnelne2 2747 |
. . . . . 6
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6 | 5 | ancoms 459 |
. . . . 5
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7 | fvfundmfvn0 5920 |
. . . . 5
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8 | df-dfat 38655 |
. . . . . 6
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9 | afvfundmfveq 38678 |
. . . . . 6
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10 | 8, 9 | sylbir 218 |
. . . . 5
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11 | eleq1 2528 |
. . . . . . 7
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12 | 11 | eqcoms 2470 |
. . . . . 6
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13 | 12 | biimpd 212 |
. . . . 5
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14 | 6, 7, 10, 13 | 4syl 19 |
. . . 4
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15 | 14 | ex 440 |
. . 3
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16 | 15 | pm2.43d 50 |
. 2
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17 | 4, 16 | impbid2 209 |
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 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-sep 4539 ax-nul 4548 ax-pow 4595 ax-pr 4653 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-nel 2636 df-ral 2754 df-rex 2755 df-rab 2758 df-v 3059 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-nul 3744 df-if 3894 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4213 df-br 4417 df-opab 4476 df-id 4768 df-xp 4859 df-rel 4860 df-cnv 4861 df-co 4862 df-dm 4863 df-res 4865 df-iota 5565 df-fun 5603 df-fv 5609 df-dfat 38655 df-afv 38656 |
This theorem is referenced by: aov0nbovbi 38735 |
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