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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 2689 | . . . 4 | |
2 | id 19 | . . . . . . 7 | |
3 | csbeq1a 3012 | . . . . . . 7 | |
4 | 2, 3 | eleq12d 2210 | . . . . . 6 |
5 | csbeq1a 3012 | . . . . . . 7 | |
6 | 5 | biantrud 302 | . . . . . 6 |
7 | 4, 6 | bitr2d 188 | . . . . 5 |
8 | 7 | equcoms 1684 | . . . 4 |
9 | 1, 8 | spcev 2780 | . . 3 |
10 | df-rex 2422 | . . . . . 6 | |
11 | nfv 1508 | . . . . . . 7 | |
12 | nfcsb1v 3035 | . . . . . . . . 9 | |
13 | 12 | nfcri 2275 | . . . . . . . 8 |
14 | nfcsb1v 3035 | . . . . . . . . 9 | |
15 | 14 | nfeq2 2293 | . . . . . . . 8 |
16 | 13, 15 | nfan 1544 | . . . . . . 7 |
17 | 5 | eqeq2d 2151 | . . . . . . . 8 |
18 | 4, 17 | anbi12d 464 | . . . . . . 7 |
19 | 11, 16, 18 | cbvex 1729 | . . . . . 6 |
20 | 10, 19 | bitri 183 | . . . . 5 |
21 | eqeq1 2146 | . . . . . . 7 | |
22 | 21 | anbi2d 459 | . . . . . 6 |
23 | 22 | exbidv 1797 | . . . . 5 |
24 | 20, 23 | syl5bb 191 | . . . 4 |
25 | rnmpt.1 | . . . . 5 | |
26 | 25 | rnmpt 4787 | . . . 4 |
27 | 24, 26 | elab2g 2831 | . . 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 1331 wex 1468 wcel 1480 wrex 2417 csb 3003 cmpt 3989 crn 4540 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-mpt 3991 df-cnv 4547 df-dm 4549 df-rn 4550 |
This theorem is referenced by: fliftel1 5695 |
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