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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 3301 | . . . . . . 7 | |
4 | 2, 3 | eqsstrri 3135 | . . . . . 6 |
5 | 4 | sseli 3098 | . . . . 5 |
6 | ffvelrn 5561 | . . . . 5 | |
7 | 1, 5, 6 | syl2an 287 | . . . 4 |
8 | off.3 | . . . . 5 | |
9 | inss2 3302 | . . . . . . 7 | |
10 | 2, 9 | eqsstrri 3135 | . . . . . 6 |
11 | 10 | sseli 3098 | . . . . 5 |
12 | ffvelrn 5561 | . . . . 5 | |
13 | 8, 11, 12 | syl2an 287 | . . . 4 |
14 | off.1 | . . . . . 6 | |
15 | 14 | ralrimivva 2517 | . . . . 5 |
16 | 15 | adantr 274 | . . . 4 |
17 | oveq1 5789 | . . . . . 6 | |
18 | 17 | eleq1d 2209 | . . . . 5 |
19 | oveq2 5790 | . . . . . 6 | |
20 | 19 | eleq1d 2209 | . . . . 5 |
21 | 18, 20 | rspc2va 2807 | . . . 4 |
22 | 7, 13, 16, 21 | syl21anc 1216 | . . 3 |
23 | eqid 2140 | . . 3 | |
24 | 22, 23 | fmptd 5582 | . 2 |
25 | ffn 5280 | . . . . 5 | |
26 | 1, 25 | syl 14 | . . . 4 |
27 | ffn 5280 | . . . . 5 | |
28 | 8, 27 | syl 14 | . . . 4 |
29 | off.4 | . . . 4 | |
30 | off.5 | . . . 4 | |
31 | eqidd 2141 | . . . 4 | |
32 | eqidd 2141 | . . . 4 | |
33 | 26, 28, 29, 30, 2, 31, 32 | offval 5997 | . . 3 |
34 | 33 | feq1d 5267 | . 2 |
35 | 24, 34 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1332 wcel 1481 wral 2417 cin 3075 cmpt 3997 wfn 5126 wf 5127 cfv 5131 (class class class)co 5782 cof 5988 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-of 5990 |
This theorem is referenced by: offeq 6003 dvaddxxbr 12873 dvmulxxbr 12874 dvaddxx 12875 dvmulxx 12876 dviaddf 12877 dvimulf 12878 |
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