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Mirrors > Home > ILE Home > Th. List > f2ndres | Unicode version |
Description: Mapping of a restriction of the (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
f2ndres |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2560 | . . . . . . . 8 | |
2 | vex 2560 | . . . . . . . 8 | |
3 | 1, 2 | op2nda 4805 | . . . . . . 7 |
4 | 3 | eleq1i 2103 | . . . . . 6 |
5 | 4 | biimpri 124 | . . . . 5 |
6 | 5 | adantl 262 | . . . 4 |
7 | 6 | rgen2 2405 | . . 3 |
8 | sneq 3386 | . . . . . . 7 | |
9 | 8 | rneqd 4563 | . . . . . 6 |
10 | 9 | unieqd 3591 | . . . . 5 |
11 | 10 | eleq1d 2106 | . . . 4 |
12 | 11 | ralxp 4479 | . . 3 |
13 | 7, 12 | mpbir 134 | . 2 |
14 | df-2nd 5768 | . . . . 5 | |
15 | 14 | reseq1i 4608 | . . . 4 |
16 | ssv 2965 | . . . . 5 | |
17 | resmpt 4656 | . . . . 5 | |
18 | 16, 17 | ax-mp 7 | . . . 4 |
19 | 15, 18 | eqtri 2060 | . . 3 |
20 | 19 | fmpt 5319 | . 2 |
21 | 13, 20 | mpbi 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1243 wcel 1393 wral 2306 cvv 2557 wss 2917 csn 3375 cop 3378 cuni 3580 cmpt 3818 cxp 4343 crn 4346 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: fo2ndresm 5789 2ndcof 5791 f2ndf 5847 |
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