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Mirrors > Home > ILE Home > Th. List > offval | Unicode version |
Description: Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.) |
Ref | Expression |
---|---|
offval.1 | |
offval.2 | |
offval.3 | |
offval.4 | |
offval.5 | |
offval.6 | |
offval.7 |
Ref | Expression |
---|---|
offval |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | offval.1 | . . . 4 | |
2 | offval.3 | . . . 4 | |
3 | fnex 5635 | . . . 4 | |
4 | 1, 2, 3 | syl2anc 408 | . . 3 |
5 | offval.2 | . . . 4 | |
6 | offval.4 | . . . 4 | |
7 | fnex 5635 | . . . 4 | |
8 | 5, 6, 7 | syl2anc 408 | . . 3 |
9 | fndm 5217 | . . . . . . . 8 | |
10 | 1, 9 | syl 14 | . . . . . . 7 |
11 | fndm 5217 | . . . . . . . 8 | |
12 | 5, 11 | syl 14 | . . . . . . 7 |
13 | 10, 12 | ineq12d 3273 | . . . . . 6 |
14 | offval.5 | . . . . . 6 | |
15 | 13, 14 | syl6eq 2186 | . . . . 5 |
16 | 15 | mpteq1d 4008 | . . . 4 |
17 | inex1g 4059 | . . . . . 6 | |
18 | 14, 17 | eqeltrrid 2225 | . . . . 5 |
19 | mptexg 5638 | . . . . 5 | |
20 | 2, 18, 19 | 3syl 17 | . . . 4 |
21 | 16, 20 | eqeltrd 2214 | . . 3 |
22 | dmeq 4734 | . . . . . 6 | |
23 | dmeq 4734 | . . . . . 6 | |
24 | 22, 23 | ineqan12d 3274 | . . . . 5 |
25 | fveq1 5413 | . . . . . 6 | |
26 | fveq1 5413 | . . . . . 6 | |
27 | 25, 26 | oveqan12d 5786 | . . . . 5 |
28 | 24, 27 | mpteq12dv 4005 | . . . 4 |
29 | df-of 5975 | . . . 4 | |
30 | 28, 29 | ovmpoga 5893 | . . 3 |
31 | 4, 8, 21, 30 | syl3anc 1216 | . 2 |
32 | 14 | eleq2i 2204 | . . . . 5 |
33 | elin 3254 | . . . . 5 | |
34 | 32, 33 | bitr3i 185 | . . . 4 |
35 | offval.6 | . . . . . 6 | |
36 | 35 | adantrr 470 | . . . . 5 |
37 | offval.7 | . . . . . 6 | |
38 | 37 | adantrl 469 | . . . . 5 |
39 | 36, 38 | oveq12d 5785 | . . . 4 |
40 | 34, 39 | sylan2b 285 | . . 3 |
41 | 40 | mpteq2dva 4013 | . 2 |
42 | 31, 16, 41 | 3eqtrd 2174 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cvv 2681 cin 3065 cmpt 3984 cdm 4534 wfn 5113 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: ofvalg 5984 off 5987 ofres 5989 offval2 5990 suppssof1 5992 ofco 5993 offveqb 5994 |
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