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Mirrors > Home > MPE Home > Th. List > fnres | Structured version Visualization version Unicode version |
Description: An equivalence for functionality of a restriction. Compare dffun8 5628. (Contributed by Mario Carneiro, 20-May-2015.) |
Ref | Expression |
---|---|
fnres |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 456 |
. . 3
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2 | vex 3060 |
. . . . . . . . . 10
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3 | 2 | brres 5130 |
. . . . . . . . 9
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4 | ancom 456 |
. . . . . . . . 9
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5 | 3, 4 | bitri 257 |
. . . . . . . 8
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6 | 5 | mobii 2333 |
. . . . . . 7
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7 | moanimv 2371 |
. . . . . . 7
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8 | 6, 7 | bitri 257 |
. . . . . 6
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9 | 8 | albii 1702 |
. . . . 5
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10 | relres 5151 |
. . . . . 6
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11 | dffun6 5616 |
. . . . . 6
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12 | 10, 11 | mpbiran 934 |
. . . . 5
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13 | df-ral 2754 |
. . . . 5
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14 | 9, 12, 13 | 3bitr4i 285 |
. . . 4
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15 | dmres 5144 |
. . . . . . 7
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16 | inss1 3664 |
. . . . . . 7
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17 | 15, 16 | eqsstri 3474 |
. . . . . 6
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18 | eqss 3459 |
. . . . . 6
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19 | 17, 18 | mpbiran 934 |
. . . . 5
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20 | dfss3 3434 |
. . . . . 6
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21 | 15 | elin2 3633 |
. . . . . . . . 9
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22 | 21 | baib 919 |
. . . . . . . 8
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23 | vex 3060 |
. . . . . . . . 9
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24 | 23 | eldm 5051 |
. . . . . . . 8
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25 | 22, 24 | syl6bb 269 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 25 | ralbiia 2830 |
. . . . . 6
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27 | 20, 26 | bitri 257 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 19, 27 | bitri 257 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 14, 28 | anbi12i 708 |
. . 3
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30 | r19.26 2929 |
. . 3
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31 | 1, 29, 30 | 3bitr4i 285 |
. 2
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32 | df-fn 5604 |
. 2
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33 | eu5 2336 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
34 | 33 | ralbii 2831 |
. 2
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35 | 31, 32, 34 | 3bitr4i 285 |
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-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-sep 4539 ax-nul 4548 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-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-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-fun 5603 df-fn 5604 |
This theorem is referenced by: f1ompt 6067 omxpenlem 7699 tz6.12-afv 38713 |
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