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Mirrors > Home > ILE Home > Th. List > mapsnconst | Unicode version |
Description: Every singleton map is a constant function. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
Ref | Expression |
---|---|
mapsncnv.s | |
mapsncnv.b | |
mapsncnv.x |
Ref | Expression |
---|---|
mapsnconst |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapsncnv.b | . . . . 5 | |
2 | mapsncnv.x | . . . . . 6 | |
3 | 2 | snex 4146 | . . . . 5 |
4 | 1, 3 | elmap 6622 | . . . 4 |
5 | 2 | fsn2 5641 | . . . . 5 |
6 | 5 | simprbi 273 | . . . 4 |
7 | 4, 6 | sylbi 120 | . . 3 |
8 | mapsncnv.s | . . . 4 | |
9 | 8 | oveq2i 5835 | . . 3 |
10 | 7, 9 | eleq2s 2252 | . 2 |
11 | 8 | xpeq1i 4606 | . . 3 |
12 | fvexg 5487 | . . . . 5 | |
13 | 2, 12 | mpan2 422 | . . . 4 |
14 | xpsng 5642 | . . . 4 | |
15 | 2, 13, 14 | sylancr 411 | . . 3 |
16 | 11, 15 | eqtr2id 2203 | . 2 |
17 | 10, 16 | eqtrd 2190 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1335 wcel 2128 cvv 2712 csn 3560 cop 3563 cxp 4584 wf 5166 cfv 5170 (class class class)co 5824 cmap 6593 |
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-in1 604 ax-in2 605 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-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 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-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-ov 5827 df-oprab 5828 df-mpo 5829 df-map 6595 |
This theorem is referenced by: mapsncnv 6640 |
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