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Mirrors > Home > ILE Home > Th. List > funresdfunsndc | Unicode version |
Description: Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in the function itself, where equality is decidable. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 30-Jan-2023.) |
Ref | Expression |
---|---|
funresdfunsndc | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funrel 5140 | . . . . 5 | |
2 | resdmdfsn 4862 | . . . . 5 | |
3 | 1, 2 | syl 14 | . . . 4 |
4 | 3 | 3ad2ant2 1003 | . . 3 DECID |
5 | 4 | uneq1d 3229 | . 2 DECID |
6 | funfn 5153 | . . 3 | |
7 | fnsnsplitdc 6401 | . . 3 DECID | |
8 | 6, 7 | syl3an2b 1253 | . 2 DECID |
9 | 5, 8 | eqtr4d 2175 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wi 4 DECID wdc 819 w3a 962 wceq 1331 wcel 1480 wral 2416 cvv 2686 cdif 3068 cun 3069 csn 3527 cop 3530 cdm 4539 cres 4541 wrel 4544 wfun 5117 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-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-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: strsetsid 11992 |
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