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Mirrors > Home > ILE Home > Th. List > off | Unicode version |
Description: The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.) |
Ref | Expression |
---|---|
off.1 | |
off.2 | |
off.3 | |
off.4 | |
off.5 | |
off.6 |
Ref | Expression |
---|---|
off |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | off.2 | . . . . 5 | |
2 | off.6 | . . . . . . 7 | |
3 | inss1 3157 | . . . . . . 7 | |
4 | 2, 3 | eqsstr3i 2976 | . . . . . 6 |
5 | 4 | sseli 2941 | . . . . 5 |
6 | ffvelrn 5300 | . . . . 5 | |
7 | 1, 5, 6 | syl2an 273 | . . . 4 |
8 | off.3 | . . . . 5 | |
9 | inss2 3158 | . . . . . . 7 | |
10 | 2, 9 | eqsstr3i 2976 | . . . . . 6 |
11 | 10 | sseli 2941 | . . . . 5 |
12 | ffvelrn 5300 | . . . . 5 | |
13 | 8, 11, 12 | syl2an 273 | . . . 4 |
14 | off.1 | . . . . . 6 | |
15 | 14 | ralrimivva 2401 | . . . . 5 |
16 | 15 | adantr 261 | . . . 4 |
17 | oveq1 5519 | . . . . . 6 | |
18 | 17 | eleq1d 2106 | . . . . 5 |
19 | oveq2 5520 | . . . . . 6 | |
20 | 19 | eleq1d 2106 | . . . . 5 |
21 | 18, 20 | rspc2va 2663 | . . . 4 |
22 | 7, 13, 16, 21 | syl21anc 1134 | . . 3 |
23 | eqid 2040 | . . 3 | |
24 | 22, 23 | fmptd 5322 | . 2 |
25 | ffn 5046 | . . . . 5 | |
26 | 1, 25 | syl 14 | . . . 4 |
27 | ffn 5046 | . . . . 5 | |
28 | 8, 27 | syl 14 | . . . 4 |
29 | off.4 | . . . 4 | |
30 | off.5 | . . . 4 | |
31 | eqidd 2041 | . . . 4 | |
32 | eqidd 2041 | . . . 4 | |
33 | 26, 28, 29, 30, 2, 31, 32 | offval 5719 | . . 3 |
34 | 33 | feq1d 5034 | . 2 |
35 | 24, 34 | mpbird 156 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 wral 2306 cin 2916 cmpt 3818 wfn 4897 wf 4898 cfv 4902 (class class class)co 5512 cof 5710 |
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-in1 544 ax-in2 545 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-coll 3872 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 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-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 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-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-of 5712 |
This theorem is referenced by: (None) |
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