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Mirrors > Home > ILE Home > Th. List > f1imass | Unicode version |
Description: Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
Ref | Expression |
---|---|
f1imass |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplrl 487 | . . . . . . 7 | |
2 | 1 | sseld 2944 | . . . . . 6 |
3 | simplr 482 | . . . . . . . . 9 | |
4 | 3 | sseld 2944 | . . . . . . . 8 |
5 | simplll 485 | . . . . . . . . 9 | |
6 | simpr 103 | . . . . . . . . 9 | |
7 | simp1rl 969 | . . . . . . . . . 10 | |
8 | 7 | 3expa 1104 | . . . . . . . . 9 |
9 | f1elima 5412 | . . . . . . . . 9 | |
10 | 5, 6, 8, 9 | syl3anc 1135 | . . . . . . . 8 |
11 | simp1rr 970 | . . . . . . . . . 10 | |
12 | 11 | 3expa 1104 | . . . . . . . . 9 |
13 | f1elima 5412 | . . . . . . . . 9 | |
14 | 5, 6, 12, 13 | syl3anc 1135 | . . . . . . . 8 |
15 | 4, 10, 14 | 3imtr3d 191 | . . . . . . 7 |
16 | 15 | ex 108 | . . . . . 6 |
17 | 2, 16 | syld 40 | . . . . 5 |
18 | 17 | pm2.43d 44 | . . . 4 |
19 | 18 | ssrdv 2951 | . . 3 |
20 | 19 | ex 108 | . 2 |
21 | imass2 4701 | . 2 | |
22 | 20, 21 | impbid1 130 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wcel 1393 wss 2917 cima 4348 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-v 2559 df-sbc 2765 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-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: f1imaeq 5414 f1imapss 5415 |
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