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Mirrors > Home > ILE Home > Th. List > ffvresb | Unicode version |
Description: A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.) |
Ref | Expression |
---|---|
ffvresb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fdm 5248 | . . . . . 6 | |
2 | dmres 4810 | . . . . . . 7 | |
3 | inss2 3267 | . . . . . . 7 | |
4 | 2, 3 | eqsstri 3099 | . . . . . 6 |
5 | 1, 4 | eqsstrrdi 3120 | . . . . 5 |
6 | 5 | sselda 3067 | . . . 4 |
7 | fvres 5413 | . . . . . 6 | |
8 | 7 | adantl 275 | . . . . 5 |
9 | ffvelrn 5521 | . . . . 5 | |
10 | 8, 9 | eqeltrrd 2195 | . . . 4 |
11 | 6, 10 | jca 304 | . . 3 |
12 | 11 | ralrimiva 2482 | . 2 |
13 | simpl 108 | . . . . . . 7 | |
14 | 13 | ralimi 2472 | . . . . . 6 |
15 | dfss3 3057 | . . . . . 6 | |
16 | 14, 15 | sylibr 133 | . . . . 5 |
17 | funfn 5123 | . . . . . 6 | |
18 | fnssres 5206 | . . . . . 6 | |
19 | 17, 18 | sylanb 282 | . . . . 5 |
20 | 16, 19 | sylan2 284 | . . . 4 |
21 | simpr 109 | . . . . . . . 8 | |
22 | 7 | eleq1d 2186 | . . . . . . . 8 |
23 | 21, 22 | syl5ibr 155 | . . . . . . 7 |
24 | 23 | ralimia 2470 | . . . . . 6 |
25 | 24 | adantl 275 | . . . . 5 |
26 | fnfvrnss 5548 | . . . . 5 | |
27 | 20, 25, 26 | syl2anc 408 | . . . 4 |
28 | df-f 5097 | . . . 4 | |
29 | 20, 27, 28 | sylanbrc 413 | . . 3 |
30 | 29 | ex 114 | . 2 |
31 | 12, 30 | impbid2 142 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 wral 2393 cin 3040 wss 3041 cdm 4509 crn 4510 cres 4511 wfun 5087 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-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-mpt 3961 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-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 |
This theorem is referenced by: resflem 5552 tfrcl 6229 frecfcllem 6269 lmbr2 12294 lmff 12329 |
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