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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 5278 | . . . . . 6 | |
2 | dmres 4840 | . . . . . . 7 | |
3 | inss2 3297 | . . . . . . 7 | |
4 | 2, 3 | eqsstri 3129 | . . . . . 6 |
5 | 1, 4 | eqsstrrdi 3150 | . . . . 5 |
6 | 5 | sselda 3097 | . . . 4 |
7 | fvres 5445 | . . . . . 6 | |
8 | 7 | adantl 275 | . . . . 5 |
9 | ffvelrn 5553 | . . . . 5 | |
10 | 8, 9 | eqeltrrd 2217 | . . . 4 |
11 | 6, 10 | jca 304 | . . 3 |
12 | 11 | ralrimiva 2505 | . 2 |
13 | simpl 108 | . . . . . . 7 | |
14 | 13 | ralimi 2495 | . . . . . 6 |
15 | dfss3 3087 | . . . . . 6 | |
16 | 14, 15 | sylibr 133 | . . . . 5 |
17 | funfn 5153 | . . . . . 6 | |
18 | fnssres 5236 | . . . . . 6 | |
19 | 17, 18 | sylanb 282 | . . . . 5 |
20 | 16, 19 | sylan2 284 | . . . 4 |
21 | simpr 109 | . . . . . . . 8 | |
22 | 7 | eleq1d 2208 | . . . . . . . 8 |
23 | 21, 22 | syl5ibr 155 | . . . . . . 7 |
24 | 23 | ralimia 2493 | . . . . . 6 |
25 | 24 | adantl 275 | . . . . 5 |
26 | fnfvrnss 5580 | . . . . 5 | |
27 | 20, 25, 26 | syl2anc 408 | . . . 4 |
28 | df-f 5127 | . . . 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 1331 wcel 1480 wral 2416 cin 3070 wss 3071 cdm 4539 crn 4540 cres 4541 wfun 5117 wfn 5118 wf 5119 cfv 5123 |
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 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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 |
This theorem is referenced by: resflem 5584 tfrcl 6261 frecfcllem 6301 lmbr2 12383 lmff 12418 |
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