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Mirrors > Home > ILE Home > Th. List > 1fv | Unicode version |
Description: A function on a singleton. (Contributed by Alexander van der Vekens, 3-Dec-2017.) |
Ref | Expression |
---|---|
1fv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 9065 | . . . . . 6 | |
2 | f1osng 5408 | . . . . . 6 | |
3 | 1, 2 | mpan 420 | . . . . 5 |
4 | f1ofo 5374 | . . . . . 6 | |
5 | dffo2 5349 | . . . . . . 7 | |
6 | 5 | biimpi 119 | . . . . . 6 |
7 | fzsn 9846 | . . . . . . . . . . . . 13 | |
8 | 1, 7 | ax-mp 5 | . . . . . . . . . . . 12 |
9 | 8 | eqcomi 2143 | . . . . . . . . . . 11 |
10 | 9 | feq2i 5266 | . . . . . . . . . 10 |
11 | 10 | biimpi 119 | . . . . . . . . 9 |
12 | snssi 3664 | . . . . . . . . 9 | |
13 | fss 5284 | . . . . . . . . 9 | |
14 | 11, 12, 13 | syl2an 287 | . . . . . . . 8 |
15 | 14 | ex 114 | . . . . . . 7 |
16 | 15 | adantr 274 | . . . . . 6 |
17 | 4, 6, 16 | 3syl 17 | . . . . 5 |
18 | 3, 17 | mpcom 36 | . . . 4 |
19 | fvsng 5616 | . . . . 5 | |
20 | 1, 19 | mpan 420 | . . . 4 |
21 | 18, 20 | jca 304 | . . 3 |
22 | 21 | adantr 274 | . 2 |
23 | feq1 5255 | . . . 4 | |
24 | fveq1 5420 | . . . . 5 | |
25 | 24 | eqeq1d 2148 | . . . 4 |
26 | 23, 25 | anbi12d 464 | . . 3 |
27 | 26 | adantl 275 | . 2 |
28 | 22, 27 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wss 3071 csn 3527 cop 3530 crn 4540 wf 5119 wfo 5121 wf1o 5122 cfv 5123 (class class class)co 5774 cc0 7620 cz 9054 cfz 9790 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 ax-rnegex 7729 ax-pre-ltirr 7732 ax-pre-apti 7735 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-neg 7936 df-z 9055 df-uz 9327 df-fz 9791 |
This theorem is referenced by: (None) |
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