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Mirrors > Home > ILE Home > Th. List > dffo4 | Unicode version |
Description: Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.) |
Ref | Expression |
---|---|
dffo4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffo2 5319 | . . 3 | |
2 | simpl 108 | . . . 4 | |
3 | vex 2663 | . . . . . . . . . 10 | |
4 | 3 | elrn 4752 | . . . . . . . . 9 |
5 | eleq2 2181 | . . . . . . . . 9 | |
6 | 4, 5 | syl5bbr 193 | . . . . . . . 8 |
7 | 6 | biimpar 295 | . . . . . . 7 |
8 | 7 | adantll 467 | . . . . . 6 |
9 | ffn 5242 | . . . . . . . . . . 11 | |
10 | fnbr 5195 | . . . . . . . . . . . 12 | |
11 | 10 | ex 114 | . . . . . . . . . . 11 |
12 | 9, 11 | syl 14 | . . . . . . . . . 10 |
13 | 12 | ancrd 324 | . . . . . . . . 9 |
14 | 13 | eximdv 1836 | . . . . . . . 8 |
15 | df-rex 2399 | . . . . . . . 8 | |
16 | 14, 15 | syl6ibr 161 | . . . . . . 7 |
17 | 16 | ad2antrr 479 | . . . . . 6 |
18 | 8, 17 | mpd 13 | . . . . 5 |
19 | 18 | ralrimiva 2482 | . . . 4 |
20 | 2, 19 | jca 304 | . . 3 |
21 | 1, 20 | sylbi 120 | . 2 |
22 | fnbrfvb 5430 | . . . . . . . . 9 | |
23 | 22 | biimprd 157 | . . . . . . . 8 |
24 | eqcom 2119 | . . . . . . . 8 | |
25 | 23, 24 | syl6ib 160 | . . . . . . 7 |
26 | 9, 25 | sylan 281 | . . . . . 6 |
27 | 26 | reximdva 2511 | . . . . 5 |
28 | 27 | ralimdv 2477 | . . . 4 |
29 | 28 | imdistani 441 | . . 3 |
30 | dffo3 5535 | . . 3 | |
31 | 29, 30 | sylibr 133 | . 2 |
32 | 21, 31 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wex 1453 wcel 1465 wral 2393 wrex 2394 class class class wbr 3899 crn 4510 wfn 5088 wf 5089 wfo 5091 cfv 5093 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fo 5099 df-fv 5101 |
This theorem is referenced by: dffo5 5537 |
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