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Mirrors > Home > ILE Home > Th. List > fnres | Unicode version |
Description: An equivalence for functionality of a restriction. Compare dffun8 5226. (Contributed by Mario Carneiro, 20-May-2015.) |
Ref | Expression |
---|---|
fnres |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 264 | . . 3 | |
2 | vex 2733 | . . . . . . . . . 10 | |
3 | 2 | brres 4897 | . . . . . . . . 9 |
4 | ancom 264 | . . . . . . . . 9 | |
5 | 3, 4 | bitri 183 | . . . . . . . 8 |
6 | 5 | mobii 2056 | . . . . . . 7 |
7 | moanimv 2094 | . . . . . . 7 | |
8 | 6, 7 | bitri 183 | . . . . . 6 |
9 | 8 | albii 1463 | . . . . 5 |
10 | relres 4919 | . . . . . 6 | |
11 | dffun6 5212 | . . . . . 6 | |
12 | 10, 11 | mpbiran 935 | . . . . 5 |
13 | df-ral 2453 | . . . . 5 | |
14 | 9, 12, 13 | 3bitr4i 211 | . . . 4 |
15 | dmres 4912 | . . . . . . 7 | |
16 | inss1 3347 | . . . . . . 7 | |
17 | 15, 16 | eqsstri 3179 | . . . . . 6 |
18 | eqss 3162 | . . . . . 6 | |
19 | 17, 18 | mpbiran 935 | . . . . 5 |
20 | dfss3 3137 | . . . . . 6 | |
21 | 15 | elin2 3315 | . . . . . . . . 9 |
22 | 21 | baib 914 | . . . . . . . 8 |
23 | vex 2733 | . . . . . . . . 9 | |
24 | 23 | eldm 4808 | . . . . . . . 8 |
25 | 22, 24 | bitrdi 195 | . . . . . . 7 |
26 | 25 | ralbiia 2484 | . . . . . 6 |
27 | 20, 26 | bitri 183 | . . . . 5 |
28 | 19, 27 | bitri 183 | . . . 4 |
29 | 14, 28 | anbi12i 457 | . . 3 |
30 | r19.26 2596 | . . 3 | |
31 | 1, 29, 30 | 3bitr4i 211 | . 2 |
32 | df-fn 5201 | . 2 | |
33 | eu5 2066 | . . 3 | |
34 | 33 | ralbii 2476 | . 2 |
35 | 31, 32, 34 | 3bitr4i 211 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1346 wceq 1348 wex 1485 weu 2019 wmo 2020 wcel 2141 wral 2448 cin 3120 wss 3121 class class class wbr 3989 cdm 4611 cres 4613 wrel 4616 wfun 5192 wfn 5193 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-res 4623 df-fun 5200 df-fn 5201 |
This theorem is referenced by: f1ompt 5647 |
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