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Mirrors > Home > ILE Home > Th. List > dom2lem | Unicode version |
Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.) |
Ref | Expression |
---|---|
dom2d.1 |
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dom2d.2 |
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Ref | Expression |
---|---|
dom2lem |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dom2d.1 |
. . . 4
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2 | 1 | ralrimiv 2549 |
. . 3
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3 | eqid 2177 |
. . . 4
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4 | 3 | fmpt 5666 |
. . 3
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5 | 2, 4 | sylib 122 |
. 2
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6 | 1 | imp 124 |
. . . . . . 7
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7 | 3 | fvmpt2 5599 |
. . . . . . . 8
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8 | 7 | adantll 476 |
. . . . . . 7
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9 | 6, 8 | mpdan 421 |
. . . . . 6
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10 | 9 | adantrr 479 |
. . . . 5
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11 | nfv 1528 |
. . . . . . . 8
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12 | nffvmpt1 5526 |
. . . . . . . . 9
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13 | 12 | nfeq1 2329 |
. . . . . . . 8
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14 | 11, 13 | nfim 1572 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | eleq1 2240 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 15 | anbi2d 464 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 16 | imbi1d 231 |
. . . . . . . 8
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18 | 15 | anbi1d 465 |
. . . . . . . . . . . 12
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19 | anidm 396 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 18, 19 | bitrdi 196 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 20 | anbi2d 464 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | fveq2 5515 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | 22 | adantr 276 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | dom2d.2 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 24 | imp 124 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 25 | biimparc 299 |
. . . . . . . . . . . 12
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27 | 23, 26 | eqeq12d 2192 |
. . . . . . . . . . 11
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28 | 27 | ex 115 |
. . . . . . . . . 10
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29 | 21, 28 | sylbird 170 |
. . . . . . . . 9
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30 | 29 | pm5.74d 182 |
. . . . . . . 8
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31 | 17, 30 | bitrd 188 |
. . . . . . 7
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32 | 14, 31, 9 | chvar 1757 |
. . . . . 6
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33 | 32 | adantrl 478 |
. . . . 5
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34 | 10, 33 | eqeq12d 2192 |
. . . 4
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35 | 25 | biimpd 144 |
. . . 4
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36 | 34, 35 | sylbid 150 |
. . 3
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37 | 36 | ralrimivva 2559 |
. 2
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38 | nfmpt1 4096 |
. . 3
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39 | nfcv 2319 |
. . 3
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40 | 38, 39 | dff13f 5770 |
. 2
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41 | 5, 37, 40 | sylanbrc 417 |
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 4121 ax-pow 4174 ax-pr 4209 |
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-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fv 5224 |
This theorem is referenced by: dom2d 6772 dom3d 6773 |
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