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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 5349 | . 2 | |
2 | ffn 5272 | . . . . 5 | |
3 | fnrnfv 5468 | . . . . . 6 | |
4 | 3 | eqeq1d 2148 | . . . . 5 |
5 | 2, 4 | syl 14 | . . . 4 |
6 | simpr 109 | . . . . . . . . . . 11 | |
7 | ffvelrn 5553 | . . . . . . . . . . . 12 | |
8 | 7 | adantr 274 | . . . . . . . . . . 11 |
9 | 6, 8 | eqeltrd 2216 | . . . . . . . . . 10 |
10 | 9 | exp31 361 | . . . . . . . . 9 |
11 | 10 | rexlimdv 2548 | . . . . . . . 8 |
12 | 11 | biantrurd 303 | . . . . . . 7 |
13 | dfbi2 385 | . . . . . . 7 | |
14 | 12, 13 | syl6rbbr 198 | . . . . . 6 |
15 | 14 | albidv 1796 | . . . . 5 |
16 | abeq1 2249 | . . . . 5 | |
17 | df-ral 2421 | . . . . 5 | |
18 | 15, 16, 17 | 3bitr4g 222 | . . . 4 |
19 | 5, 18 | bitrd 187 | . . 3 |
20 | 19 | pm5.32i 449 | . 2 |
21 | 1, 20 | bitri 183 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1329 wceq 1331 wcel 1480 cab 2125 wral 2416 wrex 2417 crn 4540 wfn 5118 wf 5119 wfo 5121 cfv 5123 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fo 5129 df-fv 5131 |
This theorem is referenced by: dffo4 5568 foco2 5655 fcofo 5685 foov 5917 0ct 6992 ctmlemr 6993 ctm 6994 ctssdclemn0 6995 ctssdccl 6996 enumctlemm 6999 cnref1o 9440 ctiunctlemfo 11952 |
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