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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 5642 | . . . 4 | |
4 | 1, 2, 3 | syl2anc 408 | . . 3 |
5 | offval.2 | . . . 4 | |
6 | offval.4 | . . . 4 | |
7 | fnex 5642 | . . . 4 | |
8 | 5, 6, 7 | syl2anc 408 | . . 3 |
9 | fndm 5222 | . . . . . . . 8 | |
10 | 1, 9 | syl 14 | . . . . . . 7 |
11 | fndm 5222 | . . . . . . . 8 | |
12 | 5, 11 | syl 14 | . . . . . . 7 |
13 | 10, 12 | ineq12d 3278 | . . . . . 6 |
14 | offval.5 | . . . . . 6 | |
15 | 13, 14 | syl6eq 2188 | . . . . 5 |
16 | 15 | mpteq1d 4013 | . . . 4 |
17 | inex1g 4064 | . . . . . 6 | |
18 | 14, 17 | eqeltrrid 2227 | . . . . 5 |
19 | mptexg 5645 | . . . . 5 | |
20 | 2, 18, 19 | 3syl 17 | . . . 4 |
21 | 16, 20 | eqeltrd 2216 | . . 3 |
22 | dmeq 4739 | . . . . . 6 | |
23 | dmeq 4739 | . . . . . 6 | |
24 | 22, 23 | ineqan12d 3279 | . . . . 5 |
25 | fveq1 5420 | . . . . . 6 | |
26 | fveq1 5420 | . . . . . 6 | |
27 | 25, 26 | oveqan12d 5793 | . . . . 5 |
28 | 24, 27 | mpteq12dv 4010 | . . . 4 |
29 | df-of 5982 | . . . 4 | |
30 | 28, 29 | ovmpoga 5900 | . . 3 |
31 | 4, 8, 21, 30 | syl3anc 1216 | . 2 |
32 | 14 | eleq2i 2206 | . . . . 5 |
33 | elin 3259 | . . . . 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 5792 | . . . 4 |
40 | 34, 39 | sylan2b 285 | . . 3 |
41 | 40 | mpteq2dva 4018 | . 2 |
42 | 31, 16, 41 | 3eqtrd 2176 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cvv 2686 cin 3070 cmpt 3989 cdm 4539 wfn 5118 cfv 5123 (class class class)co 5774 cof 5980 |
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 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-of 5982 |
This theorem is referenced by: ofvalg 5991 off 5994 ofres 5996 offval2 5997 suppssof1 5999 ofco 6000 offveqb 6001 |
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