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Mirrors > Home > ILE Home > Th. List > mpteqb | Unicode version |
Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5615. (Contributed by Mario Carneiro, 14-Nov-2014.) |
Ref | Expression |
---|---|
mpteqb |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2750 |
. . 3
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2 | 1 | ralimi 2540 |
. 2
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3 | fneq1 5306 |
. . . . . . 7
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4 | eqid 2177 |
. . . . . . . 8
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5 | 4 | mptfng 5343 |
. . . . . . 7
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6 | eqid 2177 |
. . . . . . . 8
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7 | 6 | mptfng 5343 |
. . . . . . 7
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8 | 3, 5, 7 | 3bitr4g 223 |
. . . . . 6
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9 | 8 | biimpd 144 |
. . . . 5
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10 | r19.26 2603 |
. . . . . . 7
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11 | nfmpt1 4098 |
. . . . . . . . . 10
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12 | nfmpt1 4098 |
. . . . . . . . . 10
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13 | 11, 12 | nfeq 2327 |
. . . . . . . . 9
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14 | simpll 527 |
. . . . . . . . . . . 12
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15 | 14 | fveq1d 5519 |
. . . . . . . . . . 11
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16 | 4 | fvmpt2 5601 |
. . . . . . . . . . . 12
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17 | 16 | ad2ant2lr 510 |
. . . . . . . . . . 11
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18 | 6 | fvmpt2 5601 |
. . . . . . . . . . . 12
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19 | 18 | ad2ant2l 508 |
. . . . . . . . . . 11
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20 | 15, 17, 19 | 3eqtr3d 2218 |
. . . . . . . . . 10
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21 | 20 | exp31 364 |
. . . . . . . . 9
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22 | 13, 21 | ralrimi 2548 |
. . . . . . . 8
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23 | ralim 2536 |
. . . . . . . 8
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24 | 22, 23 | syl 14 |
. . . . . . 7
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25 | 10, 24 | biimtrrid 153 |
. . . . . 6
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26 | 25 | expd 258 |
. . . . 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 2177 |
. . . 4
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30 | mpteq12 4088 |
. . . 4
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31 | 29, 30 | mpan 424 |
. . 3
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32 | 28, 31 | impbid1 142 |
. 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 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2741 df-sbc 2965 df-csb 3060 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fn 5221 df-fv 5226 |
This theorem is referenced by: eqfnfv 5615 eufnfv 5749 offveqb 6104 |
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