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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 3291 | . . . . . . 7 | |
4 | 2, 3 | eqsstrri 3125 | . . . . . 6 |
5 | 4 | sseli 3088 | . . . . 5 |
6 | ffvelrn 5546 | . . . . 5 | |
7 | 1, 5, 6 | syl2an 287 | . . . 4 |
8 | off.3 | . . . . 5 | |
9 | inss2 3292 | . . . . . . 7 | |
10 | 2, 9 | eqsstrri 3125 | . . . . . 6 |
11 | 10 | sseli 3088 | . . . . 5 |
12 | ffvelrn 5546 | . . . . 5 | |
13 | 8, 11, 12 | syl2an 287 | . . . 4 |
14 | off.1 | . . . . . 6 | |
15 | 14 | ralrimivva 2512 | . . . . 5 |
16 | 15 | adantr 274 | . . . 4 |
17 | oveq1 5774 | . . . . . 6 | |
18 | 17 | eleq1d 2206 | . . . . 5 |
19 | oveq2 5775 | . . . . . 6 | |
20 | 19 | eleq1d 2206 | . . . . 5 |
21 | 18, 20 | rspc2va 2798 | . . . 4 |
22 | 7, 13, 16, 21 | syl21anc 1215 | . . 3 |
23 | eqid 2137 | . . 3 | |
24 | 22, 23 | fmptd 5567 | . 2 |
25 | ffn 5267 | . . . . 5 | |
26 | 1, 25 | syl 14 | . . . 4 |
27 | ffn 5267 | . . . . 5 | |
28 | 8, 27 | syl 14 | . . . 4 |
29 | off.4 | . . . 4 | |
30 | off.5 | . . . 4 | |
31 | eqidd 2138 | . . . 4 | |
32 | eqidd 2138 | . . . 4 | |
33 | 26, 28, 29, 30, 2, 31, 32 | offval 5982 | . . 3 |
34 | 33 | feq1d 5254 | . 2 |
35 | 24, 34 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wral 2414 cin 3065 cmpt 3984 wfn 5113 wf 5114 cfv 5118 (class class class)co 5767 cof 5973 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-setind 4447 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-of 5975 |
This theorem is referenced by: offeq 5988 dvaddxxbr 12823 dvmulxxbr 12824 dvaddxx 12825 dvmulxx 12826 dviaddf 12827 dvimulf 12828 |
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