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Mirrors > Home > ILE Home > Th. List > fnres | Unicode version |
Description: An equivalence for functionality of a restriction. Compare dffun8 5121. (Contributed by Mario Carneiro, 20-May-2015.) |
Ref | Expression |
---|---|
fnres |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 264 | . . 3 | |
2 | vex 2663 | . . . . . . . . . 10 | |
3 | 2 | brres 4795 | . . . . . . . . 9 |
4 | ancom 264 | . . . . . . . . 9 | |
5 | 3, 4 | bitri 183 | . . . . . . . 8 |
6 | 5 | mobii 2014 | . . . . . . 7 |
7 | moanimv 2052 | . . . . . . 7 | |
8 | 6, 7 | bitri 183 | . . . . . 6 |
9 | 8 | albii 1431 | . . . . 5 |
10 | relres 4817 | . . . . . 6 | |
11 | dffun6 5107 | . . . . . 6 | |
12 | 10, 11 | mpbiran 909 | . . . . 5 |
13 | df-ral 2398 | . . . . 5 | |
14 | 9, 12, 13 | 3bitr4i 211 | . . . 4 |
15 | dmres 4810 | . . . . . . 7 | |
16 | inss1 3266 | . . . . . . 7 | |
17 | 15, 16 | eqsstri 3099 | . . . . . 6 |
18 | eqss 3082 | . . . . . 6 | |
19 | 17, 18 | mpbiran 909 | . . . . 5 |
20 | dfss3 3057 | . . . . . 6 | |
21 | 15 | elin2 3234 | . . . . . . . . 9 |
22 | 21 | baib 889 | . . . . . . . 8 |
23 | vex 2663 | . . . . . . . . 9 | |
24 | 23 | eldm 4706 | . . . . . . . 8 |
25 | 22, 24 | syl6bb 195 | . . . . . . 7 |
26 | 25 | ralbiia 2426 | . . . . . 6 |
27 | 20, 26 | bitri 183 | . . . . 5 |
28 | 19, 27 | bitri 183 | . . . 4 |
29 | 14, 28 | anbi12i 455 | . . 3 |
30 | r19.26 2535 | . . 3 | |
31 | 1, 29, 30 | 3bitr4i 211 | . 2 |
32 | df-fn 5096 | . 2 | |
33 | eu5 2024 | . . 3 | |
34 | 33 | ralbii 2418 | . 2 |
35 | 31, 32, 34 | 3bitr4i 211 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1314 wceq 1316 wex 1453 wcel 1465 weu 1977 wmo 1978 wral 2393 cin 3040 wss 3041 class class class wbr 3899 cdm 4509 cres 4511 wrel 4514 wfun 5087 wfn 5088 |
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-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 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-res 4521 df-fun 5095 df-fn 5096 |
This theorem is referenced by: f1ompt 5539 |
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