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Mirrors > Home > ILE Home > Th. List > dom3d | Unicode version |
Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by Mario Carneiro, 20-May-2013.) |
Ref | Expression |
---|---|
dom2d.1 | |
dom2d.2 | |
dom3d.3 | |
dom3d.4 |
Ref | Expression |
---|---|
dom3d |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dom2d.1 | . . . . . 6 | |
2 | dom2d.2 | . . . . . 6 | |
3 | 1, 2 | dom2lem 6252 | . . . . 5 |
4 | f1f 5092 | . . . . 5 | |
5 | 3, 4 | syl 14 | . . . 4 |
6 | dom3d.3 | . . . 4 | |
7 | dom3d.4 | . . . 4 | |
8 | fex2 5059 | . . . 4 | |
9 | 5, 6, 7, 8 | syl3anc 1135 | . . 3 |
10 | f1eq1 5087 | . . . 4 | |
11 | 10 | spcegv 2641 | . . 3 |
12 | 9, 3, 11 | sylc 56 | . 2 |
13 | brdomg 6229 | . . 3 | |
14 | 7, 13 | syl 14 | . 2 |
15 | 12, 14 | mpbird 156 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 cvv 2557 class class class wbr 3764 cmpt 3818 wf 4898 wf1 4899 cdom 6220 |
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-13 1404 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 ax-un 4170 |
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 df-dom 6223 |
This theorem is referenced by: dom3 6256 xpdom2 6305 fopwdom 6310 |
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