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Mirrors > Home > ILE Home > Th. List > f2ndf | Unicode version |
Description: The (second member of an ordered pair) function restricted to a function is a function of into the codomain of . (Contributed by Alexander van der Vekens, 4-Feb-2018.) |
Ref | Expression |
---|---|
f2ndf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f2ndres 5787 | . . 3 | |
2 | fssxp 5058 | . . 3 | |
3 | fssres 5066 | . . 3 | |
4 | 1, 2, 3 | sylancr 393 | . 2 |
5 | resabs1 4640 | . . . . 5 | |
6 | 2, 5 | syl 14 | . . . 4 |
7 | 6 | eqcomd 2045 | . . 3 |
8 | 7 | feq1d 5034 | . 2 |
9 | 4, 8 | mpbird 156 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1243 wss 2917 cxp 4343 cres 4347 wf 4898 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-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-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-fv 4910 df-2nd 5768 |
This theorem is referenced by: fo2ndf 5848 f1o2ndf1 5849 |
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