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Mirrors > Home > ILE Home > Th. List > ofres | Unicode version |
Description: Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.) |
Ref | Expression |
---|---|
ofres.1 | |
ofres.2 | |
ofres.3 | |
ofres.4 | |
ofres.5 |
Ref | Expression |
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ofres |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ofres.1 | . . 3 | |
2 | ofres.2 | . . 3 | |
3 | ofres.3 | . . 3 | |
4 | ofres.4 | . . 3 | |
5 | ofres.5 | . . 3 | |
6 | eqidd 2171 | . . 3 | |
7 | eqidd 2171 | . . 3 | |
8 | 1, 2, 3, 4, 5, 6, 7 | offval 6068 | . 2 |
9 | inss1 3347 | . . . . 5 | |
10 | 5, 9 | eqsstrri 3180 | . . . 4 |
11 | fnssres 5311 | . . . 4 | |
12 | 1, 10, 11 | sylancl 411 | . . 3 |
13 | inss2 3348 | . . . . 5 | |
14 | 5, 13 | eqsstrri 3180 | . . . 4 |
15 | fnssres 5311 | . . . 4 | |
16 | 2, 14, 15 | sylancl 411 | . . 3 |
17 | ssexg 4128 | . . . 4 | |
18 | 10, 3, 17 | sylancr 412 | . . 3 |
19 | inidm 3336 | . . 3 | |
20 | fvres 5520 | . . . 4 | |
21 | 20 | adantl 275 | . . 3 |
22 | fvres 5520 | . . . 4 | |
23 | 22 | adantl 275 | . . 3 |
24 | 12, 16, 18, 18, 19, 21, 23 | offval 6068 | . 2 |
25 | 8, 24 | eqtr4d 2206 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 cvv 2730 cin 3120 wss 3121 cmpt 4050 cres 4613 wfn 5193 cfv 5198 (class class class)co 5853 cof 6059 |
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-in1 609 ax-in2 610 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-coll 4104 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 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-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-of 6061 |
This theorem is referenced by: (None) |
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