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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 3508 | . . . . . 6 | |
2 | 1 | reseq2d 4789 | . . . . 5 |
3 | fveq2 5389 | . . . . . . 7 | |
4 | opeq12 3677 | . . . . . . 7 | |
5 | 3, 4 | mpdan 417 | . . . . . 6 |
6 | 5 | sneqd 3510 | . . . . 5 |
7 | 2, 6 | eqeq12d 2132 | . . . 4 |
8 | 7 | imbi2d 229 | . . 3 |
9 | vex 2663 | . . . . . . 7 | |
10 | 9 | snss 3619 | . . . . . 6 |
11 | fnssres 5206 | . . . . . 6 | |
12 | 10, 11 | sylan2b 285 | . . . . 5 |
13 | dffn2 5244 | . . . . . . 7 | |
14 | 9 | fsn2 5562 | . . . . . . 7 |
15 | 13, 14 | bitri 183 | . . . . . 6 |
16 | vsnid 3527 | . . . . . . . . . . 11 | |
17 | fvres 5413 | . . . . . . . . . . 11 | |
18 | 16, 17 | ax-mp 5 | . . . . . . . . . 10 |
19 | 18 | opeq2i 3679 | . . . . . . . . 9 |
20 | 19 | sneqi 3509 | . . . . . . . 8 |
21 | 20 | eqeq2i 2128 | . . . . . . 7 |
22 | snssi 3634 | . . . . . . . . . 10 | |
23 | 22, 11 | sylan2 284 | . . . . . . . . 9 |
24 | funfvex 5406 | . . . . . . . . . 10 | |
25 | 24 | funfni 5193 | . . . . . . . . 9 |
26 | 23, 16, 25 | sylancl 409 | . . . . . . . 8 |
27 | 26 | biantrurd 303 | . . . . . . 7 |
28 | 21, 27 | syl5rbbr 194 | . . . . . 6 |
29 | 15, 28 | syl5bb 191 | . . . . 5 |
30 | 12, 29 | mpbid 146 | . . . 4 |
31 | 30 | expcom 115 | . . 3 |
32 | 8, 31 | vtoclga 2726 | . 2 |
33 | 32 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 cvv 2660 wss 3041 csn 3497 cop 3500 cres 4511 wfn 5088 wf 5089 cfv 5093 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-reu 2400 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 |
This theorem is referenced by: fressnfv 5575 fnsnsplitss 5587 fnsnsplitdc 6369 dif1en 6741 fnfi 6793 fseq1p1m1 9842 resunimafz0 10542 |
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