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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 5694 | . . . 4 |
3 | eqeq2 2167 | . . . . . 6 | |
4 | 3 | bibi2d 231 | . . . . 5 |
5 | 4 | albidv 1804 | . . . 4 |
6 | 2, 5 | spcev 2807 | . . 3 |
7 | eufnfv.2 | . . . . . . 7 | |
8 | eqid 2157 | . . . . . . 7 | |
9 | 7, 8 | fnmpti 5299 | . . . . . 6 |
10 | fneq1 5259 | . . . . . 6 | |
11 | 9, 10 | mpbiri 167 | . . . . 5 |
12 | 11 | pm4.71ri 390 | . . . 4 |
13 | dffn5im 5515 | . . . . . . 7 | |
14 | 13 | eqeq1d 2166 | . . . . . 6 |
15 | funfvex 5486 | . . . . . . . . 9 | |
16 | 15 | funfni 5271 | . . . . . . . 8 |
17 | 16 | ralrimiva 2530 | . . . . . . 7 |
18 | mpteqb 5559 | . . . . . . 7 | |
19 | 17, 18 | syl 14 | . . . . . 6 |
20 | 14, 19 | bitrd 187 | . . . . 5 |
21 | 20 | pm5.32i 450 | . . . 4 |
22 | 12, 21 | bitr2i 184 | . . 3 |
23 | 6, 22 | mpg 1431 | . 2 |
24 | df-eu 2009 | . 2 | |
25 | 23, 24 | mpbir 145 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wal 1333 wceq 1335 wex 1472 weu 2006 wcel 2128 wral 2435 cvv 2712 cmpt 4026 wfn 5166 cfv 5171 |
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-coll 4080 ax-sep 4083 ax-pow 4136 ax-pr 4170 |
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-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 |
This theorem is referenced by: (None) |
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