Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ofvalg | Unicode version |
Description: Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Jim Kingdon, 22-Nov-2023.) |
Ref | Expression |
---|---|
offval.1 | |
offval.2 | |
offval.3 | |
offval.4 | |
offval.5 | |
ofval.6 | |
ofval.7 | |
ofval.8 |
Ref | Expression |
---|---|
ofvalg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | offval.1 | . . . . 5 | |
2 | offval.2 | . . . . 5 | |
3 | offval.3 | . . . . 5 | |
4 | offval.4 | . . . . 5 | |
5 | offval.5 | . . . . 5 | |
6 | eqidd 2165 | . . . . 5 | |
7 | eqidd 2165 | . . . . 5 | |
8 | 1, 2, 3, 4, 5, 6, 7 | offval 6051 | . . . 4 |
9 | 8 | fveq1d 5482 | . . 3 |
10 | 9 | adantr 274 | . 2 |
11 | eqid 2164 | . . 3 | |
12 | fveq2 5480 | . . . 4 | |
13 | fveq2 5480 | . . . 4 | |
14 | 12, 13 | oveq12d 5854 | . . 3 |
15 | simpr 109 | . . 3 | |
16 | inss1 3337 | . . . . . . . 8 | |
17 | 5, 16 | eqsstrri 3170 | . . . . . . 7 |
18 | 17 | sseli 3133 | . . . . . 6 |
19 | ofval.6 | . . . . . 6 | |
20 | 18, 19 | sylan2 284 | . . . . 5 |
21 | inss2 3338 | . . . . . . . 8 | |
22 | 5, 21 | eqsstrri 3170 | . . . . . . 7 |
23 | 22 | sseli 3133 | . . . . . 6 |
24 | ofval.7 | . . . . . 6 | |
25 | 23, 24 | sylan2 284 | . . . . 5 |
26 | 20, 25 | oveq12d 5854 | . . . 4 |
27 | ofval.8 | . . . 4 | |
28 | 26, 27 | eqeltrd 2241 | . . 3 |
29 | 11, 14, 15, 28 | fvmptd3 5573 | . 2 |
30 | 10, 29, 26 | 3eqtrd 2201 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 cin 3110 cmpt 4037 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: offeq 6057 dvaddxxbr 13212 dvmulxxbr 13213 |
Copyright terms: Public domain | W3C validator |