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Mirrors > Home > ILE Home > Th. List > elrnmptg | Unicode version |
Description: Membership in the range of a function. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
rnmpt.1 |
Ref | Expression |
---|---|
elrnmptg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnmpt.1 | . . . 4 | |
2 | 1 | rnmpt 4757 | . . 3 |
3 | 2 | eleq2i 2184 | . 2 |
4 | r19.29 2546 | . . . . 5 | |
5 | eleq1 2180 | . . . . . . . 8 | |
6 | 5 | biimparc 297 | . . . . . . 7 |
7 | elex 2671 | . . . . . . 7 | |
8 | 6, 7 | syl 14 | . . . . . 6 |
9 | 8 | rexlimivw 2522 | . . . . 5 |
10 | 4, 9 | syl 14 | . . . 4 |
11 | 10 | ex 114 | . . 3 |
12 | eqeq1 2124 | . . . . 5 | |
13 | 12 | rexbidv 2415 | . . . 4 |
14 | 13 | elab3g 2808 | . . 3 |
15 | 11, 14 | syl 14 | . 2 |
16 | 3, 15 | syl5bb 191 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 cab 2103 wral 2393 wrex 2394 cvv 2660 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-ral 2398 df-rex 2399 df-v 2662 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: elrnmpti 4762 fliftel 5662 |
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