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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 | |
dom2d.2 |
Ref | Expression |
---|---|
dom2lem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dom2d.1 | . . . 4 | |
2 | 1 | ralrimiv 2481 | . . 3 |
3 | eqid 2117 | . . . 4 | |
4 | 3 | fmpt 5538 | . . 3 |
5 | 2, 4 | sylib 121 | . 2 |
6 | 1 | imp 123 | . . . . . . 7 |
7 | 3 | fvmpt2 5472 | . . . . . . . 8 |
8 | 7 | adantll 467 | . . . . . . 7 |
9 | 6, 8 | mpdan 417 | . . . . . 6 |
10 | 9 | adantrr 470 | . . . . 5 |
11 | nfv 1493 | . . . . . . . 8 | |
12 | nffvmpt1 5400 | . . . . . . . . 9 | |
13 | 12 | nfeq1 2268 | . . . . . . . 8 |
14 | 11, 13 | nfim 1536 | . . . . . . 7 |
15 | eleq1 2180 | . . . . . . . . . 10 | |
16 | 15 | anbi2d 459 | . . . . . . . . 9 |
17 | 16 | imbi1d 230 | . . . . . . . 8 |
18 | 15 | anbi1d 460 | . . . . . . . . . . . 12 |
19 | anidm 393 | . . . . . . . . . . . 12 | |
20 | 18, 19 | syl6bb 195 | . . . . . . . . . . 11 |
21 | 20 | anbi2d 459 | . . . . . . . . . 10 |
22 | fveq2 5389 | . . . . . . . . . . . . 13 | |
23 | 22 | adantr 274 | . . . . . . . . . . . 12 |
24 | dom2d.2 | . . . . . . . . . . . . . 14 | |
25 | 24 | imp 123 | . . . . . . . . . . . . 13 |
26 | 25 | biimparc 297 | . . . . . . . . . . . 12 |
27 | 23, 26 | eqeq12d 2132 | . . . . . . . . . . 11 |
28 | 27 | ex 114 | . . . . . . . . . 10 |
29 | 21, 28 | sylbird 169 | . . . . . . . . 9 |
30 | 29 | pm5.74d 181 | . . . . . . . 8 |
31 | 17, 30 | bitrd 187 | . . . . . . 7 |
32 | 14, 31, 9 | chvar 1715 | . . . . . 6 |
33 | 32 | adantrl 469 | . . . . 5 |
34 | 10, 33 | eqeq12d 2132 | . . . 4 |
35 | 25 | biimpd 143 | . . . 4 |
36 | 34, 35 | sylbid 149 | . . 3 |
37 | 36 | ralrimivva 2491 | . 2 |
38 | nfmpt1 3991 | . . 3 | |
39 | nfcv 2258 | . . 3 | |
40 | 38, 39 | dff13f 5639 | . 2 |
41 | 5, 37, 40 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 wral 2393 cmpt 3959 wf 5089 wf1 5090 cfv 5093 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fv 5101 |
This theorem is referenced by: dom2d 6635 dom3d 6636 |
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