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Mirrors > Home > MPE Home > Th. List > mpteqb | Structured version Visualization version Unicode version |
Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5991. (Contributed by Mario Carneiro, 14-Nov-2014.) |
Ref | Expression |
---|---|
mpteqb |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3040 |
. . 3
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2 | 1 | ralimi 2796 |
. 2
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3 | fneq1 5674 |
. . . . . . 7
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4 | eqid 2471 |
. . . . . . . 8
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5 | 4 | mptfng 5713 |
. . . . . . 7
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6 | eqid 2471 |
. . . . . . . 8
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7 | 6 | mptfng 5713 |
. . . . . . 7
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8 | 3, 5, 7 | 3bitr4g 296 |
. . . . . 6
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9 | 8 | biimpd 212 |
. . . . 5
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10 | r19.26 2904 |
. . . . . . 7
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11 | nfmpt1 4485 |
. . . . . . . . . 10
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12 | nfmpt1 4485 |
. . . . . . . . . 10
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13 | 11, 12 | nfeq 2623 |
. . . . . . . . 9
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14 | simpll 768 |
. . . . . . . . . . . 12
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15 | 14 | fveq1d 5881 |
. . . . . . . . . . 11
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16 | 4 | fvmpt2 5972 |
. . . . . . . . . . . 12
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17 | 16 | ad2ant2lr 762 |
. . . . . . . . . . 11
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18 | 6 | fvmpt2 5972 |
. . . . . . . . . . . 12
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19 | 18 | ad2ant2l 760 |
. . . . . . . . . . 11
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20 | 15, 17, 19 | 3eqtr3d 2513 |
. . . . . . . . . 10
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21 | 20 | exp31 615 |
. . . . . . . . 9
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22 | 13, 21 | ralrimi 2800 |
. . . . . . . 8
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23 | ralim 2792 |
. . . . . . . 8
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24 | 22, 23 | syl 17 |
. . . . . . 7
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25 | 10, 24 | syl5bir 226 |
. . . . . 6
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26 | 25 | expd 443 |
. . . . 5
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27 | 9, 26 | mpdd 40 |
. . . 4
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28 | 27 | com12 31 |
. . 3
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29 | eqid 2471 |
. . . 4
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30 | mpteq12 4475 |
. . . 4
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31 | 29, 30 | mpan 684 |
. . 3
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32 | 28, 31 | impbid1 208 |
. 2
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33 | 2, 32 | syl 17 |
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-fn 5592 df-fv 5597 |
This theorem is referenced by: eqfnfv 5991 eufnfv 6156 offveqb 6572 ramcl 15066 fucsect 15955 setcepi 16061 0frgp 17507 dprdf11 17734 dpjeq 17770 mvrf1 18726 mplmonmul 18765 frgpcyg 19221 ustuqtop 21339 mdegle0 23105 ply1nzb 23150 cvmliftphtlem 30112 |
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