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Mirrors > Home > ILE Home > Th. List > resflem | Unicode version |
Description: A lemma to bound the range of a restriction. The conclusion would also hold with in place of (provided does not occur in ). If that stronger result is needed, it is however simpler to use the instance of resflem 5631 where is substituted for (in both the conclusion and the third hypothesis). (Contributed by BJ, 4-Jul-2022.) |
Ref | Expression |
---|---|
resflem.1 | |
resflem.2 | |
resflem.3 |
Ref | Expression |
---|---|
resflem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resflem.2 | . . . . . 6 | |
2 | 1 | sseld 3127 | . . . . 5 |
3 | resflem.1 | . . . . . . 7 | |
4 | fdm 5325 | . . . . . . 7 | |
5 | 3, 4 | syl 14 | . . . . . 6 |
6 | 5 | eleq2d 2227 | . . . . 5 |
7 | 2, 6 | sylibrd 168 | . . . 4 |
8 | resflem.3 | . . . . 5 | |
9 | 8 | ex 114 | . . . 4 |
10 | 7, 9 | jcad 305 | . . 3 |
11 | 10 | ralrimiv 2529 | . 2 |
12 | ffun 5322 | . . . 4 | |
13 | 3, 12 | syl 14 | . . 3 |
14 | ffvresb 5630 | . . 3 | |
15 | 13, 14 | syl 14 | . 2 |
16 | 11, 15 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wcel 2128 wral 2435 wss 3102 cdm 4586 cres 4588 wfun 5164 wf 5166 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-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-v 2714 df-sbc 2938 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-mpt 4027 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-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-fv 5178 |
This theorem is referenced by: bj-charfun 13393 |
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