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Mirrors > Home > ILE Home > Th. List > elrnmpt2 | Unicode version |
Description: Membership in the range of an operation class abstraction. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
rngop.1 | |
elrnmpt2.1 |
Ref | Expression |
---|---|
elrnmpt2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngop.1 | . . . 4 | |
2 | 1 | rnmpt2 5611 | . . 3 |
3 | 2 | eleq2i 2104 | . 2 |
4 | elrnmpt2.1 | . . . . . 6 | |
5 | eleq1 2100 | . . . . . 6 | |
6 | 4, 5 | mpbiri 157 | . . . . 5 |
7 | 6 | rexlimivw 2429 | . . . 4 |
8 | 7 | rexlimivw 2429 | . . 3 |
9 | eqeq1 2046 | . . . 4 | |
10 | 9 | 2rexbidv 2349 | . . 3 |
11 | 8, 10 | elab3 2694 | . 2 |
12 | 3, 11 | bitri 173 | 1 |
Colors of variables: wff set class |
Syntax hints: wb 98 wceq 1243 wcel 1393 cab 2026 wrex 2307 cvv 2557 crn 4346 cmpt2 5514 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-cnv 4353 df-dm 4355 df-rn 4356 df-oprab 5516 df-mpt2 5517 |
This theorem is referenced by: (None) |
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