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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 2604 | . . . 4 | |
2 | id 19 | . . . . . . 7 | |
3 | csbeq1a 2916 | . . . . . . 7 | |
4 | 2, 3 | eleq12d 2149 | . . . . . 6 |
5 | csbeq1a 2916 | . . . . . . 7 | |
6 | 5 | biantrud 298 | . . . . . 6 |
7 | 4, 6 | bitr2d 187 | . . . . 5 |
8 | 7 | equcoms 1634 | . . . 4 |
9 | 1, 8 | spcev 2692 | . . 3 |
10 | df-rex 2354 | . . . . . 6 | |
11 | nfv 1461 | . . . . . . 7 | |
12 | nfcsb1v 2938 | . . . . . . . . 9 | |
13 | 12 | nfcri 2213 | . . . . . . . 8 |
14 | nfcsb1v 2938 | . . . . . . . . 9 | |
15 | 14 | nfeq2 2230 | . . . . . . . 8 |
16 | 13, 15 | nfan 1497 | . . . . . . 7 |
17 | 5 | eqeq2d 2092 | . . . . . . . 8 |
18 | 4, 17 | anbi12d 456 | . . . . . . 7 |
19 | 11, 16, 18 | cbvex 1679 | . . . . . 6 |
20 | 10, 19 | bitri 182 | . . . . 5 |
21 | eqeq1 2087 | . . . . . . 7 | |
22 | 21 | anbi2d 451 | . . . . . 6 |
23 | 22 | exbidv 1746 | . . . . 5 |
24 | 20, 23 | syl5bb 190 | . . . 4 |
25 | rnmpt.1 | . . . . 5 | |
26 | 25 | rnmpt 4600 | . . . 4 |
27 | 24, 26 | elab2g 2740 | . . 3 |
28 | 9, 27 | syl5ibr 154 | . 2 |
29 | 28 | impcom 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wex 1421 wcel 1433 wrex 2349 csb 2908 cmpt 3839 crn 4364 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-mpt 3841 df-cnv 4371 df-dm 4373 df-rn 4374 |
This theorem is referenced by: fliftel1 5454 |
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