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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 4834 | . . 3 |
3 | 2 | eleq2i 2224 | . 2 |
4 | r19.29 2594 | . . . . 5 | |
5 | eleq1 2220 | . . . . . . . 8 | |
6 | 5 | biimparc 297 | . . . . . . 7 |
7 | elex 2723 | . . . . . . 7 | |
8 | 6, 7 | syl 14 | . . . . . 6 |
9 | 8 | rexlimivw 2570 | . . . . 5 |
10 | 4, 9 | syl 14 | . . . 4 |
11 | 10 | ex 114 | . . 3 |
12 | eqeq1 2164 | . . . . 5 | |
13 | 12 | rexbidv 2458 | . . . 4 |
14 | 13 | elab3g 2863 | . . 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 1335 wcel 2128 cab 2143 wral 2435 wrex 2436 cvv 2712 cmpt 4025 crn 4587 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-br 3966 df-opab 4026 df-mpt 4027 df-cnv 4594 df-dm 4596 df-rn 4597 |
This theorem is referenced by: elrnmpti 4839 fliftel 5743 |
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