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Mirrors > Home > ILE Home > Th. List > f1elima | Unicode version |
Description: Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
f1elima |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1fn 5300 | . . . 4 | |
2 | fvelimab 5445 | . . . 4 | |
3 | 1, 2 | sylan 281 | . . 3 |
4 | 3 | 3adant2 985 | . 2 |
5 | ssel 3061 | . . . . . . . 8 | |
6 | 5 | impac 378 | . . . . . . 7 |
7 | f1fveq 5641 | . . . . . . . . . . . 12 | |
8 | 7 | ancom2s 540 | . . . . . . . . . . 11 |
9 | 8 | biimpd 143 | . . . . . . . . . 10 |
10 | 9 | anassrs 397 | . . . . . . . . 9 |
11 | eleq1 2180 | . . . . . . . . . 10 | |
12 | 11 | biimpcd 158 | . . . . . . . . 9 |
13 | 10, 12 | sylan9 406 | . . . . . . . 8 |
14 | 13 | anasss 396 | . . . . . . 7 |
15 | 6, 14 | sylan2 284 | . . . . . 6 |
16 | 15 | anassrs 397 | . . . . 5 |
17 | 16 | rexlimdva 2526 | . . . 4 |
18 | 17 | 3impa 1161 | . . 3 |
19 | eqid 2117 | . . . 4 | |
20 | fveq2 5389 | . . . . . 6 | |
21 | 20 | eqeq1d 2126 | . . . . 5 |
22 | 21 | rspcev 2763 | . . . 4 |
23 | 19, 22 | mpan2 421 | . . 3 |
24 | 18, 23 | impbid1 141 | . 2 |
25 | 4, 24 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 947 wceq 1316 wcel 1465 wrex 2394 wss 3041 cima 4512 wfn 5088 wf1 5090 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-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fv 5101 |
This theorem is referenced by: f1imass 5643 iseqf1olemnab 10229 ctinfom 11868 |
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