Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > offval2 | Unicode version |
Description: The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.) |
Ref | Expression |
---|---|
offval2.1 | |
offval2.2 | |
offval2.3 | |
offval2.4 | |
offval2.5 |
Ref | Expression |
---|---|
offval2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | offval2.2 | . . . . . 6 | |
2 | 1 | ralrimiva 2503 | . . . . 5 |
3 | eqid 2137 | . . . . . 6 | |
4 | 3 | fnmpt 5244 | . . . . 5 |
5 | 2, 4 | syl 14 | . . . 4 |
6 | offval2.4 | . . . . 5 | |
7 | 6 | fneq1d 5208 | . . . 4 |
8 | 5, 7 | mpbird 166 | . . 3 |
9 | offval2.3 | . . . . . 6 | |
10 | 9 | ralrimiva 2503 | . . . . 5 |
11 | eqid 2137 | . . . . . 6 | |
12 | 11 | fnmpt 5244 | . . . . 5 |
13 | 10, 12 | syl 14 | . . . 4 |
14 | offval2.5 | . . . . 5 | |
15 | 14 | fneq1d 5208 | . . . 4 |
16 | 13, 15 | mpbird 166 | . . 3 |
17 | offval2.1 | . . 3 | |
18 | inidm 3280 | . . 3 | |
19 | 6 | adantr 274 | . . . 4 |
20 | 19 | fveq1d 5416 | . . 3 |
21 | 14 | adantr 274 | . . . 4 |
22 | 21 | fveq1d 5416 | . . 3 |
23 | 8, 16, 17, 17, 18, 20, 22 | offval 5982 | . 2 |
24 | nffvmpt1 5425 | . . . . 5 | |
25 | nfcv 2279 | . . . . 5 | |
26 | nffvmpt1 5425 | . . . . 5 | |
27 | 24, 25, 26 | nfov 5794 | . . . 4 |
28 | nfcv 2279 | . . . 4 | |
29 | fveq2 5414 | . . . . 5 | |
30 | fveq2 5414 | . . . . 5 | |
31 | 29, 30 | oveq12d 5785 | . . . 4 |
32 | 27, 28, 31 | cbvmpt 4018 | . . 3 |
33 | simpr 109 | . . . . . 6 | |
34 | 3 | fvmpt2 5497 | . . . . . 6 |
35 | 33, 1, 34 | syl2anc 408 | . . . . 5 |
36 | 11 | fvmpt2 5497 | . . . . . 6 |
37 | 33, 9, 36 | syl2anc 408 | . . . . 5 |
38 | 35, 37 | oveq12d 5785 | . . . 4 |
39 | 38 | mpteq2dva 4013 | . . 3 |
40 | 32, 39 | syl5eq 2182 | . 2 |
41 | 23, 40 | eqtrd 2170 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wral 2414 cmpt 3984 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: ofc12 5995 caofinvl 5997 caofcom 5998 dvimulf 12828 dvexp 12833 dvmptaddx 12839 dvmptmulx 12840 dvef 12845 |
Copyright terms: Public domain | W3C validator |