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Mirrors > Home > ILE Home > Th. List > mpt2fvex | Unicode version |
Description: Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.) |
Ref | Expression |
---|---|
fmpt2.1 |
Ref | Expression |
---|---|
mpt2fvex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 5515 | . 2 | |
2 | elex 2566 | . . . . . . . . 9 | |
3 | 2 | alimi 1344 | . . . . . . . 8 |
4 | vex 2560 | . . . . . . . . 9 | |
5 | 2ndexg 5795 | . . . . . . . . 9 | |
6 | nfcv 2178 | . . . . . . . . . 10 | |
7 | nfcsb1v 2882 | . . . . . . . . . . 11 | |
8 | 7 | nfel1 2188 | . . . . . . . . . 10 |
9 | csbeq1a 2860 | . . . . . . . . . . 11 | |
10 | 9 | eleq1d 2106 | . . . . . . . . . 10 |
11 | 6, 8, 10 | spcgf 2635 | . . . . . . . . 9 |
12 | 4, 5, 11 | mp2b 8 | . . . . . . . 8 |
13 | 3, 12 | syl 14 | . . . . . . 7 |
14 | 13 | alimi 1344 | . . . . . 6 |
15 | 1stexg 5794 | . . . . . . 7 | |
16 | nfcv 2178 | . . . . . . . 8 | |
17 | nfcsb1v 2882 | . . . . . . . . 9 | |
18 | 17 | nfel1 2188 | . . . . . . . 8 |
19 | csbeq1a 2860 | . . . . . . . . 9 | |
20 | 19 | eleq1d 2106 | . . . . . . . 8 |
21 | 16, 18, 20 | spcgf 2635 | . . . . . . 7 |
22 | 4, 15, 21 | mp2b 8 | . . . . . 6 |
23 | 14, 22 | syl 14 | . . . . 5 |
24 | 23 | alrimiv 1754 | . . . 4 |
25 | 24 | 3ad2ant1 925 | . . 3 |
26 | opexg 3964 | . . . 4 | |
27 | 26 | 3adant1 922 | . . 3 |
28 | fmpt2.1 | . . . . 5 | |
29 | mpt2mptsx 5823 | . . . . 5 | |
30 | 28, 29 | eqtri 2060 | . . . 4 |
31 | 30 | mptfvex 5256 | . . 3 |
32 | 25, 27, 31 | syl2anc 391 | . 2 |
33 | 1, 32 | syl5eqel 2124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 w3a 885 wal 1241 wceq 1243 wcel 1393 cvv 2557 csb 2852 csn 3375 cop 3378 ciun 3657 cmpt 3818 cxp 4343 cfv 4902 (class class class)co 5512 cmpt2 5514 c1st 5765 c2nd 5766 |
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-13 1404 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 ax-un 4170 |
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-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fo 4908 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 |
This theorem is referenced by: mpt2fvexi 5832 oaexg 6028 omexg 6031 oeiexg 6033 |
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