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Mirrors > Home > ILE Home > Th. List > dffo3 | Unicode version |
Description: An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.) |
Ref | Expression |
---|---|
dffo3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffo2 5130 | . 2 | |
2 | ffn 5066 | . . . . 5 | |
3 | fnrnfv 5241 | . . . . . 6 | |
4 | 3 | eqeq1d 2089 | . . . . 5 |
5 | 2, 4 | syl 14 | . . . 4 |
6 | simpr 108 | . . . . . . . . . . 11 | |
7 | ffvelrn 5321 | . . . . . . . . . . . 12 | |
8 | 7 | adantr 270 | . . . . . . . . . . 11 |
9 | 6, 8 | eqeltrd 2155 | . . . . . . . . . 10 |
10 | 9 | exp31 356 | . . . . . . . . 9 |
11 | 10 | rexlimdv 2476 | . . . . . . . 8 |
12 | 11 | biantrurd 299 | . . . . . . 7 |
13 | dfbi2 380 | . . . . . . 7 | |
14 | 12, 13 | syl6rbbr 197 | . . . . . 6 |
15 | 14 | albidv 1745 | . . . . 5 |
16 | abeq1 2188 | . . . . 5 | |
17 | df-ral 2353 | . . . . 5 | |
18 | 15, 16, 17 | 3bitr4g 221 | . . . 4 |
19 | 5, 18 | bitrd 186 | . . 3 |
20 | 19 | pm5.32i 441 | . 2 |
21 | 1, 20 | bitri 182 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wal 1282 wceq 1284 wcel 1433 cab 2067 wral 2348 wrex 2349 crn 4364 wfn 4917 wf 4918 wfo 4920 cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fo 4928 df-fv 4930 |
This theorem is referenced by: dffo4 5336 foelrn 5338 foco2 5339 fcofo 5444 foov 5667 cnref1o 8733 |
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