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Mirrors > Home > ILE Home > Th. List > fnsnsplitdc | Unicode version |
Description: Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 29-Jan-2023.) |
Ref | Expression |
---|---|
fnsnsplitdc | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnresdm 5232 | . . 3 | |
2 | 1 | 3ad2ant2 1003 | . 2 DECID |
3 | resundi 4832 | . . 3 | |
4 | dcdifsnid 6400 | . . . . 5 DECID | |
5 | 4 | 3adant2 1000 | . . . 4 DECID |
6 | 5 | reseq2d 4819 | . . 3 DECID |
7 | fnressn 5606 | . . . . 5 | |
8 | 7 | uneq2d 3230 | . . . 4 |
9 | 8 | 3adant1 999 | . . 3 DECID |
10 | 3, 6, 9 | 3eqtr3a 2196 | . 2 DECID |
11 | 2, 10 | eqtr3d 2174 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 DECID wdc 819 w3a 962 wceq 1331 wcel 1480 wral 2416 cdif 3068 cun 3069 csn 3527 cop 3530 cres 4541 wfn 5118 cfv 5123 |
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-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 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3an 964 df-tru 1334 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-ral 2421 df-rex 2422 df-reu 2423 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-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 |
This theorem is referenced by: funresdfunsndc 6402 |
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