Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > fnressn | Unicode version |
Description: A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.) |
Ref | Expression |
---|---|
fnressn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 3409 | . . . . . 6 | |
2 | 1 | reseq2d 4630 | . . . . 5 |
3 | fveq2 5198 | . . . . . . 7 | |
4 | opeq12 3572 | . . . . . . 7 | |
5 | 3, 4 | mpdan 412 | . . . . . 6 |
6 | 5 | sneqd 3411 | . . . . 5 |
7 | 2, 6 | eqeq12d 2095 | . . . 4 |
8 | 7 | imbi2d 228 | . . 3 |
9 | vex 2604 | . . . . . . 7 | |
10 | 9 | snss 3516 | . . . . . 6 |
11 | fnssres 5032 | . . . . . 6 | |
12 | 10, 11 | sylan2b 281 | . . . . 5 |
13 | dffn2 5067 | . . . . . . 7 | |
14 | 9 | fsn2 5358 | . . . . . . 7 |
15 | 13, 14 | bitri 182 | . . . . . 6 |
16 | vsnid 3426 | . . . . . . . . . . 11 | |
17 | fvres 5219 | . . . . . . . . . . 11 | |
18 | 16, 17 | ax-mp 7 | . . . . . . . . . 10 |
19 | 18 | opeq2i 3574 | . . . . . . . . 9 |
20 | 19 | sneqi 3410 | . . . . . . . 8 |
21 | 20 | eqeq2i 2091 | . . . . . . 7 |
22 | snssi 3529 | . . . . . . . . . 10 | |
23 | 22, 11 | sylan2 280 | . . . . . . . . 9 |
24 | funfvex 5212 | . . . . . . . . . 10 | |
25 | 24 | funfni 5019 | . . . . . . . . 9 |
26 | 23, 16, 25 | sylancl 404 | . . . . . . . 8 |
27 | 26 | biantrurd 299 | . . . . . . 7 |
28 | 21, 27 | syl5rbbr 193 | . . . . . 6 |
29 | 15, 28 | syl5bb 190 | . . . . 5 |
30 | 12, 29 | mpbid 145 | . . . 4 |
31 | 30 | expcom 114 | . . 3 |
32 | 8, 31 | vtoclga 2664 | . 2 |
33 | 32 | impcom 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 cvv 2601 wss 2973 csn 3398 cop 3401 cres 4365 wfn 4917 wf 4918 cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-reu 2355 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 |
This theorem is referenced by: fressnfv 5371 dif1en 6364 fnfi 6388 fseq1p1m1 9111 |
Copyright terms: Public domain | W3C validator |