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Mirrors > Home > ILE Home > Th. List > mpteqb | Unicode version |
Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5526. (Contributed by Mario Carneiro, 14-Nov-2014.) |
Ref | Expression |
---|---|
mpteqb |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2700 |
. . 3
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2 | 1 | ralimi 2498 |
. 2
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3 | fneq1 5219 |
. . . . . . 7
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4 | eqid 2140 |
. . . . . . . 8
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5 | 4 | mptfng 5256 |
. . . . . . 7
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6 | eqid 2140 |
. . . . . . . 8
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7 | 6 | mptfng 5256 |
. . . . . . 7
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8 | 3, 5, 7 | 3bitr4g 222 |
. . . . . 6
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9 | 8 | biimpd 143 |
. . . . 5
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10 | r19.26 2561 |
. . . . . . 7
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11 | nfmpt1 4029 |
. . . . . . . . . 10
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12 | nfmpt1 4029 |
. . . . . . . . . 10
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13 | 11, 12 | nfeq 2290 |
. . . . . . . . 9
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14 | simpll 519 |
. . . . . . . . . . . 12
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15 | 14 | fveq1d 5431 |
. . . . . . . . . . 11
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16 | 4 | fvmpt2 5512 |
. . . . . . . . . . . 12
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17 | 16 | ad2ant2lr 502 |
. . . . . . . . . . 11
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18 | 6 | fvmpt2 5512 |
. . . . . . . . . . . 12
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19 | 18 | ad2ant2l 500 |
. . . . . . . . . . 11
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20 | 15, 17, 19 | 3eqtr3d 2181 |
. . . . . . . . . 10
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21 | 20 | exp31 362 |
. . . . . . . . 9
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22 | 13, 21 | ralrimi 2506 |
. . . . . . . 8
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23 | ralim 2494 |
. . . . . . . 8
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24 | 22, 23 | syl 14 |
. . . . . . 7
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25 | 10, 24 | syl5bir 152 |
. . . . . 6
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26 | 25 | expd 256 |
. . . . 5
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27 | 9, 26 | mpdd 41 |
. . . 4
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28 | 27 | com12 30 |
. . 3
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29 | eqid 2140 |
. . . 4
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30 | mpteq12 4019 |
. . . 4
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31 | 29, 30 | mpan 421 |
. . 3
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32 | 28, 31 | impbid1 141 |
. 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-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-csb 3008 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fn 5134 df-fv 5139 |
This theorem is referenced by: eqfnfv 5526 eufnfv 5656 offveqb 6009 |
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