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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 5325 | . . . 4 | |
2 | fvelimab 5470 | . . . 4 | |
3 | 1, 2 | sylan 281 | . . 3 |
4 | 3 | 3adant2 1000 | . 2 |
5 | ssel 3086 | . . . . . . . 8 | |
6 | 5 | impac 378 | . . . . . . 7 |
7 | f1fveq 5666 | . . . . . . . . . . . 12 | |
8 | 7 | ancom2s 555 | . . . . . . . . . . 11 |
9 | 8 | biimpd 143 | . . . . . . . . . 10 |
10 | 9 | anassrs 397 | . . . . . . . . 9 |
11 | eleq1 2200 | . . . . . . . . . 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 2547 | . . . 4 |
18 | 17 | 3impa 1176 | . . 3 |
19 | eqid 2137 | . . . 4 | |
20 | fveq2 5414 | . . . . . 6 | |
21 | 20 | eqeq1d 2146 | . . . . 5 |
22 | 21 | rspcev 2784 | . . . 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 962 wceq 1331 wcel 1480 wrex 2415 wss 3066 cima 4537 wfn 5113 wf1 5115 cfv 5118 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fv 5126 |
This theorem is referenced by: f1imass 5668 iseqf1olemnab 10254 ctinfom 11930 |
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