Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4600 | . . 3 |
3 | 2 | eleq2i 2145 | . 2 |
4 | r19.29 2494 | . . . . 5 | |
5 | eleq1 2141 | . . . . . . . 8 | |
6 | 5 | biimparc 293 | . . . . . . 7 |
7 | elex 2610 | . . . . . . 7 | |
8 | 6, 7 | syl 14 | . . . . . 6 |
9 | 8 | rexlimivw 2473 | . . . . 5 |
10 | 4, 9 | syl 14 | . . . 4 |
11 | 10 | ex 113 | . . 3 |
12 | eqeq1 2087 | . . . . 5 | |
13 | 12 | rexbidv 2369 | . . . 4 |
14 | 13 | elab3g 2744 | . . 3 |
15 | 11, 14 | syl 14 | . 2 |
16 | 3, 15 | syl5bb 190 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 cab 2067 wral 2348 wrex 2349 cvv 2601 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-ral 2353 df-rex 2354 df-v 2603 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: elrnmpti 4605 fliftel 5453 |
Copyright terms: Public domain | W3C validator |