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Mirrors > Home > ILE Home > Th. List > mpteqb | Unicode version |
Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5208. (Contributed by Mario Carneiro, 14-Nov-2014.) |
Ref | Expression |
---|---|
mpteqb |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2560 |
. . 3
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2 | 1 | ralimi 2378 |
. 2
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3 | fneq1 4930 |
. . . . . . 7
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4 | eqid 2037 |
. . . . . . . 8
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5 | 4 | mptfng 4967 |
. . . . . . 7
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6 | eqid 2037 |
. . . . . . . 8
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7 | 6 | mptfng 4967 |
. . . . . . 7
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8 | 3, 5, 7 | 3bitr4g 212 |
. . . . . 6
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9 | 8 | biimpd 132 |
. . . . 5
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10 | r19.26 2435 |
. . . . . . 7
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11 | nfmpt1 3841 |
. . . . . . . . . 10
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12 | nfmpt1 3841 |
. . . . . . . . . 10
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13 | 11, 12 | nfeq 2182 |
. . . . . . . . 9
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14 | simpll 481 |
. . . . . . . . . . . 12
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15 | 14 | fveq1d 5123 |
. . . . . . . . . . 11
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16 | 4 | fvmpt2 5197 |
. . . . . . . . . . . 12
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17 | 16 | ad2ant2lr 479 |
. . . . . . . . . . 11
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18 | 6 | fvmpt2 5197 |
. . . . . . . . . . . 12
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19 | 18 | ad2ant2l 477 |
. . . . . . . . . . 11
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20 | 15, 17, 19 | 3eqtr3d 2077 |
. . . . . . . . . 10
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21 | 20 | exp31 346 |
. . . . . . . . 9
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22 | 13, 21 | ralrimi 2384 |
. . . . . . . 8
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23 | ralim 2374 |
. . . . . . . 8
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24 | 22, 23 | syl 14 |
. . . . . . 7
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25 | 10, 24 | syl5bir 142 |
. . . . . 6
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26 | 25 | expd 245 |
. . . . 5
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27 | 9, 26 | mpdd 36 |
. . . 4
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28 | 27 | com12 27 |
. . 3
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29 | eqid 2037 |
. . . 4
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30 | mpteq12 3831 |
. . . 4
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31 | 29, 30 | mpan 400 |
. . 3
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32 | 28, 31 | impbid1 130 |
. 2
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33 | 2, 32 | syl 14 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-iota 4810 df-fun 4847 df-fn 4848 df-fv 4853 |
This theorem is referenced by: eqfnfv 5208 eufnfv 5332 offveqb 5672 |
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