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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 5396 | . 2 | |
2 | ffn 5319 | . . . . 5 | |
3 | fnrnfv 5515 | . . . . . 6 | |
4 | 3 | eqeq1d 2166 | . . . . 5 |
5 | 2, 4 | syl 14 | . . . 4 |
6 | dfbi2 386 | . . . . . . 7 | |
7 | simpr 109 | . . . . . . . . . . 11 | |
8 | ffvelrn 5600 | . . . . . . . . . . . 12 | |
9 | 8 | adantr 274 | . . . . . . . . . . 11 |
10 | 7, 9 | eqeltrd 2234 | . . . . . . . . . 10 |
11 | 10 | exp31 362 | . . . . . . . . 9 |
12 | 11 | rexlimdv 2573 | . . . . . . . 8 |
13 | 12 | biantrurd 303 | . . . . . . 7 |
14 | 6, 13 | bitr4id 198 | . . . . . 6 |
15 | 14 | albidv 1804 | . . . . 5 |
16 | abeq1 2267 | . . . . 5 | |
17 | df-ral 2440 | . . . . 5 | |
18 | 15, 16, 17 | 3bitr4g 222 | . . . 4 |
19 | 5, 18 | bitrd 187 | . . 3 |
20 | 19 | pm5.32i 450 | . 2 |
21 | 1, 20 | bitri 183 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1333 wceq 1335 wcel 2128 cab 2143 wral 2435 wrex 2436 crn 4587 wfn 5165 wf 5166 wfo 5168 cfv 5170 |
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-sep 4082 ax-pow 4135 ax-pr 4169 |
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-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-fo 5176 df-fv 5178 |
This theorem is referenced by: dffo4 5615 foco2 5704 fcofo 5734 foov 5967 0ct 7051 ctmlemr 7052 ctm 7053 ctssdclemn0 7054 ctssdccl 7055 enumctlemm 7058 cnref1o 9559 ctiunctlemfo 12179 ioocosf1o 13186 |
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