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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 4787 | . . 3 |
3 | 2 | eleq2i 2206 | . 2 |
4 | r19.29 2569 | . . . . 5 | |
5 | eleq1 2202 | . . . . . . . 8 | |
6 | 5 | biimparc 297 | . . . . . . 7 |
7 | elex 2697 | . . . . . . 7 | |
8 | 6, 7 | syl 14 | . . . . . 6 |
9 | 8 | rexlimivw 2545 | . . . . 5 |
10 | 4, 9 | syl 14 | . . . 4 |
11 | 10 | ex 114 | . . 3 |
12 | eqeq1 2146 | . . . . 5 | |
13 | 12 | rexbidv 2438 | . . . 4 |
14 | 13 | elab3g 2835 | . . 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 1331 wcel 1480 cab 2125 wral 2416 wrex 2417 cvv 2686 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-ral 2421 df-rex 2422 df-v 2688 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: elrnmpti 4792 fliftel 5694 |
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