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Mirrors > Home > ILE Home > Th. List > elrnmpt1 | Unicode version |
Description: Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
rnmpt.1 |
Ref | Expression |
---|---|
elrnmpt1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2663 | . . . 4 | |
2 | id 19 | . . . . . . 7 | |
3 | csbeq1a 2983 | . . . . . . 7 | |
4 | 2, 3 | eleq12d 2188 | . . . . . 6 |
5 | csbeq1a 2983 | . . . . . . 7 | |
6 | 5 | biantrud 302 | . . . . . 6 |
7 | 4, 6 | bitr2d 188 | . . . . 5 |
8 | 7 | equcoms 1669 | . . . 4 |
9 | 1, 8 | spcev 2754 | . . 3 |
10 | df-rex 2399 | . . . . . 6 | |
11 | nfv 1493 | . . . . . . 7 | |
12 | nfcsb1v 3005 | . . . . . . . . 9 | |
13 | 12 | nfcri 2252 | . . . . . . . 8 |
14 | nfcsb1v 3005 | . . . . . . . . 9 | |
15 | 14 | nfeq2 2270 | . . . . . . . 8 |
16 | 13, 15 | nfan 1529 | . . . . . . 7 |
17 | 5 | eqeq2d 2129 | . . . . . . . 8 |
18 | 4, 17 | anbi12d 464 | . . . . . . 7 |
19 | 11, 16, 18 | cbvex 1714 | . . . . . 6 |
20 | 10, 19 | bitri 183 | . . . . 5 |
21 | eqeq1 2124 | . . . . . . 7 | |
22 | 21 | anbi2d 459 | . . . . . 6 |
23 | 22 | exbidv 1781 | . . . . 5 |
24 | 20, 23 | syl5bb 191 | . . . 4 |
25 | rnmpt.1 | . . . . 5 | |
26 | 25 | rnmpt 4757 | . . . 4 |
27 | 24, 26 | elab2g 2804 | . . 3 |
28 | 9, 27 | syl5ibr 155 | . 2 |
29 | 28 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wex 1453 wcel 1465 wrex 2394 csb 2975 cmpt 3959 crn 4510 |
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-rex 2399 df-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-mpt 3961 df-cnv 4517 df-dm 4519 df-rn 4520 |
This theorem is referenced by: fliftel1 5663 |
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