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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 5701 | . . . 4 | |
4 | 1, 2, 3 | syl2anc 409 | . . 3 |
5 | offval.2 | . . . 4 | |
6 | offval.4 | . . . 4 | |
7 | fnex 5701 | . . . 4 | |
8 | 5, 6, 7 | syl2anc 409 | . . 3 |
9 | fndm 5281 | . . . . . . . 8 | |
10 | 1, 9 | syl 14 | . . . . . . 7 |
11 | fndm 5281 | . . . . . . . 8 | |
12 | 5, 11 | syl 14 | . . . . . . 7 |
13 | 10, 12 | ineq12d 3319 | . . . . . 6 |
14 | offval.5 | . . . . . 6 | |
15 | 13, 14 | eqtrdi 2213 | . . . . 5 |
16 | 15 | mpteq1d 4061 | . . . 4 |
17 | inex1g 4112 | . . . . . 6 | |
18 | 14, 17 | eqeltrrid 2252 | . . . . 5 |
19 | mptexg 5704 | . . . . 5 | |
20 | 2, 18, 19 | 3syl 17 | . . . 4 |
21 | 16, 20 | eqeltrd 2241 | . . 3 |
22 | dmeq 4798 | . . . . . 6 | |
23 | dmeq 4798 | . . . . . 6 | |
24 | 22, 23 | ineqan12d 3320 | . . . . 5 |
25 | fveq1 5479 | . . . . . 6 | |
26 | fveq1 5479 | . . . . . 6 | |
27 | 25, 26 | oveqan12d 5855 | . . . . 5 |
28 | 24, 27 | mpteq12dv 4058 | . . . 4 |
29 | df-of 6044 | . . . 4 | |
30 | 28, 29 | ovmpoga 5962 | . . 3 |
31 | 4, 8, 21, 30 | syl3anc 1227 | . 2 |
32 | 14 | eleq2i 2231 | . . . . 5 |
33 | elin 3300 | . . . . 5 | |
34 | 32, 33 | bitr3i 185 | . . . 4 |
35 | offval.6 | . . . . . 6 | |
36 | 35 | adantrr 471 | . . . . 5 |
37 | offval.7 | . . . . . 6 | |
38 | 37 | adantrl 470 | . . . . 5 |
39 | 36, 38 | oveq12d 5854 | . . . 4 |
40 | 34, 39 | sylan2b 285 | . . 3 |
41 | 40 | mpteq2dva 4066 | . 2 |
42 | 31, 16, 41 | 3eqtrd 2201 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 cvv 2721 cin 3110 cmpt 4037 cdm 4598 wfn 5177 cfv 5182 (class class class)co 5836 cof 6042 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-of 6044 |
This theorem is referenced by: ofvalg 6053 off 6056 ofres 6058 offval2 6059 suppssof1 6061 ofco 6062 offveqb 6063 |
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