Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 | |
dom2d.2 |
Ref | Expression |
---|---|
dom2lem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dom2d.1 | . . . 4 | |
2 | 1 | ralrimiv 2391 | . . 3 |
3 | eqid 2040 | . . . 4 | |
4 | 3 | fmpt 5319 | . . 3 |
5 | 2, 4 | sylib 127 | . 2 |
6 | 1 | imp 115 | . . . . . . 7 |
7 | 3 | fvmpt2 5254 | . . . . . . . 8 |
8 | 7 | adantll 445 | . . . . . . 7 |
9 | 6, 8 | mpdan 398 | . . . . . 6 |
10 | 9 | adantrr 448 | . . . . 5 |
11 | nfv 1421 | . . . . . . . 8 | |
12 | nffvmpt1 5186 | . . . . . . . . 9 | |
13 | 12 | nfeq1 2187 | . . . . . . . 8 |
14 | 11, 13 | nfim 1464 | . . . . . . 7 |
15 | eleq1 2100 | . . . . . . . . . 10 | |
16 | 15 | anbi2d 437 | . . . . . . . . 9 |
17 | 16 | imbi1d 220 | . . . . . . . 8 |
18 | 15 | anbi1d 438 | . . . . . . . . . . . 12 |
19 | anidm 376 | . . . . . . . . . . . 12 | |
20 | 18, 19 | syl6bb 185 | . . . . . . . . . . 11 |
21 | 20 | anbi2d 437 | . . . . . . . . . 10 |
22 | fveq2 5178 | . . . . . . . . . . . . 13 | |
23 | 22 | adantr 261 | . . . . . . . . . . . 12 |
24 | dom2d.2 | . . . . . . . . . . . . . 14 | |
25 | 24 | imp 115 | . . . . . . . . . . . . 13 |
26 | 25 | biimparc 283 | . . . . . . . . . . . 12 |
27 | 23, 26 | eqeq12d 2054 | . . . . . . . . . . 11 |
28 | 27 | ex 108 | . . . . . . . . . 10 |
29 | 21, 28 | sylbird 159 | . . . . . . . . 9 |
30 | 29 | pm5.74d 171 | . . . . . . . 8 |
31 | 17, 30 | bitrd 177 | . . . . . . 7 |
32 | 14, 31, 9 | chvar 1640 | . . . . . 6 |
33 | 32 | adantrl 447 | . . . . 5 |
34 | 10, 33 | eqeq12d 2054 | . . . 4 |
35 | 25 | biimpd 132 | . . . 4 |
36 | 34, 35 | sylbid 139 | . . 3 |
37 | 36 | ralrimivva 2401 | . 2 |
38 | nfmpt1 3850 | . . 3 | |
39 | nfcv 2178 | . . 3 | |
40 | 38, 39 | dff13f 5409 | . 2 |
41 | 5, 37, 40 | sylanbrc 394 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wral 2306 cmpt 3818 wf 4898 wf1 4899 cfv 4902 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fv 4910 |
This theorem is referenced by: dom2d 6253 dom3d 6254 |
Copyright terms: Public domain | W3C validator |