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Mirrors > Home > ILE Home > Th. List > fo2ndf | Unicode version |
Description: The (second member of an ordered pair) function restricted to a function is a function of onto the range of . (Contributed by Alexander van der Vekens, 4-Feb-2018.) |
Ref | Expression |
---|---|
fo2ndf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 5046 | . . . 4 | |
2 | dffn3 5053 | . . . 4 | |
3 | 1, 2 | sylib 127 | . . 3 |
4 | f2ndf 5847 | . . 3 | |
5 | 3, 4 | syl 14 | . 2 |
6 | 2, 4 | sylbi 114 | . . . . 5 |
7 | 1, 6 | syl 14 | . . . 4 |
8 | frn 5052 | . . . 4 | |
9 | 7, 8 | syl 14 | . . 3 |
10 | elrn2g 4525 | . . . . . 6 | |
11 | 10 | ibi 165 | . . . . 5 |
12 | fvres 5198 | . . . . . . . . . 10 | |
13 | 12 | adantl 262 | . . . . . . . . 9 |
14 | vex 2560 | . . . . . . . . . 10 | |
15 | vex 2560 | . . . . . . . . . 10 | |
16 | 14, 15 | op2nd 5774 | . . . . . . . . 9 |
17 | 13, 16 | syl6req 2089 | . . . . . . . 8 |
18 | f2ndf 5847 | . . . . . . . . . 10 | |
19 | ffn 5046 | . . . . . . . . . 10 | |
20 | 18, 19 | syl 14 | . . . . . . . . 9 |
21 | fnfvelrn 5299 | . . . . . . . . 9 | |
22 | 20, 21 | sylan 267 | . . . . . . . 8 |
23 | 17, 22 | eqeltrd 2114 | . . . . . . 7 |
24 | 23 | ex 108 | . . . . . 6 |
25 | 24 | exlimdv 1700 | . . . . 5 |
26 | 11, 25 | syl5 28 | . . . 4 |
27 | 26 | ssrdv 2951 | . . 3 |
28 | 9, 27 | eqssd 2962 | . 2 |
29 | dffo2 5110 | . 2 | |
30 | 5, 28, 29 | sylanbrc 394 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wex 1381 wcel 1393 wss 2917 cop 3378 crn 4346 cres 4347 wfn 4897 wf 4898 wfo 4900 cfv 4902 c2nd 5766 |
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-iun 3659 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-fo 4908 df-fv 4910 df-2nd 5768 |
This theorem is referenced by: f1o2ndf1 5849 |
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