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Mirrors > Home > NFE Home > Th. List > fvfullfunlem3 | Unicode version |
Description: Lemma for fvfullfun 5864. Part one of the full function definition agrees with the set itself over its domain. (Contributed by SF, 9-Mar-2015.) |
Ref | Expression |
---|---|
fvfullfunlem3 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffun2 5119 |
. . . 4
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2 | brres 4949 |
. . . . . . 7
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3 | fvfullfunlem1 5861 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | 3 | abeq2i 2460 |
. . . . . . . 8
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5 | 4 | anbi2i 675 |
. . . . . . 7
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6 | 2, 5 | bitri 240 |
. . . . . 6
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7 | brres 4949 |
. . . . . . 7
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8 | fvfullfunlem1 5861 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 8 | abeq2i 2460 |
. . . . . . . 8
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10 | 9 | anbi2i 675 |
. . . . . . 7
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11 | 7, 10 | bitri 240 |
. . . . . 6
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12 | tz6.12-1 5344 |
. . . . . . . . 9
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13 | 12 | adantrl 696 |
. . . . . . . 8
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14 | tz6.12-1 5344 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 14 | adantl 452 |
. . . . . . . 8
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16 | 13, 15 | eqtr3d 2387 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 16 | adantlr 695 |
. . . . . 6
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18 | 6, 11, 17 | syl2anb 465 |
. . . . 5
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19 | 18 | gen2 1547 |
. . . 4
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20 | 1, 19 | mpgbir 1550 |
. . 3
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21 | fvfullfunlem2 5862 |
. . . . 5
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22 | ssdmrn 5099 |
. . . . . 6
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23 | ssv 3291 |
. . . . . . 7
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24 | xpss2 4857 |
. . . . . . 7
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25 | 23, 24 | ax-mp 5 |
. . . . . 6
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26 | 22, 25 | sstri 3281 |
. . . . 5
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27 | 21, 26 | ssini 3478 |
. . . 4
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28 | df-res 4788 |
. . . 4
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29 | 27, 28 | sseqtr4i 3304 |
. . 3
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30 | funssfv 5343 |
. . 3
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31 | 20, 29, 30 | mp3an12 1267 |
. 2
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32 | fvres 5342 |
. 2
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33 | 31, 32 | eqtr3d 2387 |
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 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-co 4726 df-ima 4727 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fv 4795 |
This theorem is referenced by: fvfullfun 5864 |
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