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Mirrors > Home > ILE Home > Th. List > funresdfunsnss | 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 a subset of the function itself. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 21-Jan-2023.) |
Ref | Expression |
---|---|
funresdfunsnss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funrel 5187 | . . . . 5 | |
2 | resdmdfsn 4909 | . . . . 5 | |
3 | 1, 2 | syl 14 | . . . 4 |
4 | 3 | adantr 274 | . . 3 |
5 | 4 | uneq1d 3260 | . 2 |
6 | funfn 5200 | . . 3 | |
7 | fnsnsplitss 5666 | . . 3 | |
8 | 6, 7 | sylanb 282 | . 2 |
9 | 5, 8 | eqsstrd 3164 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 cvv 2712 cdif 3099 cun 3100 wss 3102 csn 3560 cop 3563 cdm 4586 cres 4588 wrel 4591 wfun 5164 wfn 5165 cfv 5170 |
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-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 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-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 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-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 |
This theorem is referenced by: (None) |
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