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Mirrors > Home > ILE Home > Th. List > eufnfv | Unicode version |
Description: A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.) |
Ref | Expression |
---|---|
eufnfv.1 | |
eufnfv.2 |
Ref | Expression |
---|---|
eufnfv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eufnfv.1 | . . . . 5 | |
2 | 1 | mptex 5722 | . . . 4 |
3 | eqeq2 2180 | . . . . . 6 | |
4 | 3 | bibi2d 231 | . . . . 5 |
5 | 4 | albidv 1817 | . . . 4 |
6 | 2, 5 | spcev 2825 | . . 3 |
7 | eufnfv.2 | . . . . . . 7 | |
8 | eqid 2170 | . . . . . . 7 | |
9 | 7, 8 | fnmpti 5326 | . . . . . 6 |
10 | fneq1 5286 | . . . . . 6 | |
11 | 9, 10 | mpbiri 167 | . . . . 5 |
12 | 11 | pm4.71ri 390 | . . . 4 |
13 | dffn5im 5542 | . . . . . . 7 | |
14 | 13 | eqeq1d 2179 | . . . . . 6 |
15 | funfvex 5513 | . . . . . . . . 9 | |
16 | 15 | funfni 5298 | . . . . . . . 8 |
17 | 16 | ralrimiva 2543 | . . . . . . 7 |
18 | mpteqb 5586 | . . . . . . 7 | |
19 | 17, 18 | syl 14 | . . . . . 6 |
20 | 14, 19 | bitrd 187 | . . . . 5 |
21 | 20 | pm5.32i 451 | . . . 4 |
22 | 12, 21 | bitr2i 184 | . . 3 |
23 | 6, 22 | mpg 1444 | . 2 |
24 | df-eu 2022 | . 2 | |
25 | 23, 24 | mpbir 145 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wal 1346 wceq 1348 wex 1485 weu 2019 wcel 2141 wral 2448 cvv 2730 cmpt 4050 wfn 5193 cfv 5198 |
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-coll 4104 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-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 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 |
This theorem is referenced by: (None) |
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