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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 5660 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 3146 | . . . . 5 |
3 | resflem.1 | . . . . . . 7 | |
4 | fdm 5353 | . . . . . . 7 | |
5 | 3, 4 | syl 14 | . . . . . 6 |
6 | 5 | eleq2d 2240 | . . . . 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 2542 | . 2 |
12 | ffun 5350 | . . . 4 | |
13 | 3, 12 | syl 14 | . . 3 |
14 | ffvresb 5659 | . . 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 1348 wcel 2141 wral 2448 wss 3121 cdm 4611 cres 4613 wfun 5192 wf 5194 cfv 5198 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 |
This theorem is referenced by: bj-charfun 13842 |
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