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Mirrors > Home > ILE Home > Th. List > fnres | Unicode version |
Description: An equivalence for functionality of a restriction. Compare dffun8 5198. (Contributed by Mario Carneiro, 20-May-2015.) |
Ref | Expression |
---|---|
fnres |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 264 | . . 3 | |
2 | vex 2715 | . . . . . . . . . 10 | |
3 | 2 | brres 4872 | . . . . . . . . 9 |
4 | ancom 264 | . . . . . . . . 9 | |
5 | 3, 4 | bitri 183 | . . . . . . . 8 |
6 | 5 | mobii 2043 | . . . . . . 7 |
7 | moanimv 2081 | . . . . . . 7 | |
8 | 6, 7 | bitri 183 | . . . . . 6 |
9 | 8 | albii 1450 | . . . . 5 |
10 | relres 4894 | . . . . . 6 | |
11 | dffun6 5184 | . . . . . 6 | |
12 | 10, 11 | mpbiran 925 | . . . . 5 |
13 | df-ral 2440 | . . . . 5 | |
14 | 9, 12, 13 | 3bitr4i 211 | . . . 4 |
15 | dmres 4887 | . . . . . . 7 | |
16 | inss1 3327 | . . . . . . 7 | |
17 | 15, 16 | eqsstri 3160 | . . . . . 6 |
18 | eqss 3143 | . . . . . 6 | |
19 | 17, 18 | mpbiran 925 | . . . . 5 |
20 | dfss3 3118 | . . . . . 6 | |
21 | 15 | elin2 3295 | . . . . . . . . 9 |
22 | 21 | baib 905 | . . . . . . . 8 |
23 | vex 2715 | . . . . . . . . 9 | |
24 | 23 | eldm 4783 | . . . . . . . 8 |
25 | 22, 24 | bitrdi 195 | . . . . . . 7 |
26 | 25 | ralbiia 2471 | . . . . . 6 |
27 | 20, 26 | bitri 183 | . . . . 5 |
28 | 19, 27 | bitri 183 | . . . 4 |
29 | 14, 28 | anbi12i 456 | . . 3 |
30 | r19.26 2583 | . . 3 | |
31 | 1, 29, 30 | 3bitr4i 211 | . 2 |
32 | df-fn 5173 | . 2 | |
33 | eu5 2053 | . . 3 | |
34 | 33 | ralbii 2463 | . 2 |
35 | 31, 32, 34 | 3bitr4i 211 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1333 wceq 1335 wex 1472 weu 2006 wmo 2007 wcel 2128 wral 2435 cin 3101 wss 3102 class class class wbr 3965 cdm 4586 cres 4588 wrel 4591 wfun 5164 wfn 5165 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-res 4598 df-fun 5172 df-fn 5173 |
This theorem is referenced by: f1ompt 5618 |
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