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Mirrors > Home > MPE Home > Th. List > dom2lem | Structured version Visualization version 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 2965 | . . 3 |
3 | eqid 2622 | . . . 4 | |
4 | 3 | fmpt 6381 | . . 3 |
5 | 2, 4 | sylib 208 | . 2 |
6 | 1 | imp 445 | . . . . . . 7 |
7 | 3 | fvmpt2 6291 | . . . . . . . 8 |
8 | 7 | adantll 750 | . . . . . . 7 |
9 | 6, 8 | mpdan 702 | . . . . . 6 |
10 | 9 | adantrr 753 | . . . . 5 |
11 | nfv 1843 | . . . . . . . 8 | |
12 | nffvmpt1 6199 | . . . . . . . . 9 | |
13 | 12 | nfeq1 2778 | . . . . . . . 8 |
14 | 11, 13 | nfim 1825 | . . . . . . 7 |
15 | eleq1 2689 | . . . . . . . . . 10 | |
16 | 15 | anbi2d 740 | . . . . . . . . 9 |
17 | 16 | imbi1d 331 | . . . . . . . 8 |
18 | 15 | anbi1d 741 | . . . . . . . . . . . 12 |
19 | anidm 676 | . . . . . . . . . . . 12 | |
20 | 18, 19 | syl6bb 276 | . . . . . . . . . . 11 |
21 | 20 | anbi2d 740 | . . . . . . . . . 10 |
22 | fveq2 6191 | . . . . . . . . . . . . 13 | |
23 | 22 | adantr 481 | . . . . . . . . . . . 12 |
24 | dom2d.2 | . . . . . . . . . . . . . 14 | |
25 | 24 | imp 445 | . . . . . . . . . . . . 13 |
26 | 25 | biimparc 504 | . . . . . . . . . . . 12 |
27 | 23, 26 | eqeq12d 2637 | . . . . . . . . . . 11 |
28 | 27 | ex 450 | . . . . . . . . . 10 |
29 | 21, 28 | sylbird 250 | . . . . . . . . 9 |
30 | 29 | pm5.74d 262 | . . . . . . . 8 |
31 | 17, 30 | bitrd 268 | . . . . . . 7 |
32 | 14, 31, 9 | chvar 2262 | . . . . . 6 |
33 | 32 | adantrl 752 | . . . . 5 |
34 | 10, 33 | eqeq12d 2637 | . . . 4 |
35 | 25 | biimpd 219 | . . . 4 |
36 | 34, 35 | sylbid 230 | . . 3 |
37 | 36 | ralrimivva 2971 | . 2 |
38 | nfmpt1 4747 | . . 3 | |
39 | nfcv 2764 | . . 3 | |
40 | 38, 39 | dff13f 6513 | . 2 |
41 | 5, 37, 40 | sylanbrc 698 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 cmpt 4729 wf 5884 wf1 5885 cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fv 5896 |
This theorem is referenced by: dom2d 7996 dom3d 7997 ixpfi2 8264 infxpenc2lem1 8842 dfac12lem2 8966 4sqlem11 15659 odf1o1 17987 odf1o2 17988 dis2ndc 21263 hauspwpwf1 21791 itg1addlem4 23466 basellem3 24809 fsumvma 24938 dchrisum0fno1 25200 |
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