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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 4171 | . . . . 5 |
4 | 1, 3 | elmap 6655 | . . . 4 |
5 | 2 | fsn2 5670 | . . . . 5 |
6 | 5 | simprbi 273 | . . . 4 |
7 | 4, 6 | sylbi 120 | . . 3 |
8 | mapsncnv.s | . . . 4 | |
9 | 8 | oveq2i 5864 | . . 3 |
10 | 7, 9 | eleq2s 2265 | . 2 |
11 | 8 | xpeq1i 4631 | . . 3 |
12 | fvexg 5515 | . . . . 5 | |
13 | 2, 12 | mpan2 423 | . . . 4 |
14 | xpsng 5671 | . . . 4 | |
15 | 2, 13, 14 | sylancr 412 | . . 3 |
16 | 11, 15 | eqtr2id 2216 | . 2 |
17 | 10, 16 | eqtrd 2203 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1348 wcel 2141 cvv 2730 csn 3583 cop 3586 cxp 4609 wf 5194 cfv 5198 (class class class)co 5853 cmap 6626 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-map 6628 |
This theorem is referenced by: mapsncnv 6673 |
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