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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 5424 | . 2 | |
2 | ffn 5347 | . . . . 5 | |
3 | fnrnfv 5543 | . . . . . 6 | |
4 | 3 | eqeq1d 2179 | . . . . 5 |
5 | 2, 4 | syl 14 | . . . 4 |
6 | dfbi2 386 | . . . . . . 7 | |
7 | simpr 109 | . . . . . . . . . . 11 | |
8 | ffvelrn 5629 | . . . . . . . . . . . 12 | |
9 | 8 | adantr 274 | . . . . . . . . . . 11 |
10 | 7, 9 | eqeltrd 2247 | . . . . . . . . . 10 |
11 | 10 | exp31 362 | . . . . . . . . 9 |
12 | 11 | rexlimdv 2586 | . . . . . . . 8 |
13 | 12 | biantrurd 303 | . . . . . . 7 |
14 | 6, 13 | bitr4id 198 | . . . . . 6 |
15 | 14 | albidv 1817 | . . . . 5 |
16 | abeq1 2280 | . . . . 5 | |
17 | df-ral 2453 | . . . . 5 | |
18 | 15, 16, 17 | 3bitr4g 222 | . . . 4 |
19 | 5, 18 | bitrd 187 | . . 3 |
20 | 19 | pm5.32i 451 | . 2 |
21 | 1, 20 | bitri 183 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1346 wceq 1348 wcel 2141 cab 2156 wral 2448 wrex 2449 crn 4612 wfn 5193 wf 5194 wfo 5196 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-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-v 2732 df-sbc 2956 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-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-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fo 5204 df-fv 5206 |
This theorem is referenced by: dffo4 5644 foco2 5733 fcofo 5763 foov 5999 0ct 7084 ctmlemr 7085 ctm 7086 ctssdclemn0 7087 ctssdccl 7088 enumctlemm 7091 cnref1o 9609 1arith 12319 ctiunctlemfo 12394 ioocosf1o 13569 |
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