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Mirrors > Home > ILE Home > Th. List > isoselem | Unicode version |
Description: Lemma for isose 5715. (Contributed by Mario Carneiro, 23-Jun-2015.) |
Ref | Expression |
---|---|
isofrlem.1 | |
isofrlem.2 |
Ref | Expression |
---|---|
isoselem | Se Se |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfse2 4907 | . . . . . . . . 9 Se | |
2 | 1 | biimpi 119 | . . . . . . . 8 Se |
3 | 2 | r19.21bi 2518 | . . . . . . 7 Se |
4 | 3 | expcom 115 | . . . . . 6 Se |
5 | 4 | adantl 275 | . . . . 5 Se |
6 | imaeq2 4872 | . . . . . . . . . . 11 | |
7 | 6 | eleq1d 2206 | . . . . . . . . . 10 |
8 | 7 | imbi2d 229 | . . . . . . . . 9 |
9 | isofrlem.2 | . . . . . . . . 9 | |
10 | 8, 9 | vtoclg 2741 | . . . . . . . 8 |
11 | 10 | com12 30 | . . . . . . 7 |
12 | 11 | adantr 274 | . . . . . 6 |
13 | isofrlem.1 | . . . . . . . 8 | |
14 | isoini 5712 | . . . . . . . 8 | |
15 | 13, 14 | sylan 281 | . . . . . . 7 |
16 | 15 | eleq1d 2206 | . . . . . 6 |
17 | 12, 16 | sylibd 148 | . . . . 5 |
18 | 5, 17 | syld 45 | . . . 4 Se |
19 | 18 | ralrimdva 2510 | . . 3 Se |
20 | isof1o 5701 | . . . . 5 | |
21 | f1ofn 5361 | . . . . 5 | |
22 | sneq 3533 | . . . . . . . . 9 | |
23 | 22 | imaeq2d 4876 | . . . . . . . 8 |
24 | 23 | ineq2d 3272 | . . . . . . 7 |
25 | 24 | eleq1d 2206 | . . . . . 6 |
26 | 25 | ralrn 5551 | . . . . 5 |
27 | 13, 20, 21, 26 | 4syl 18 | . . . 4 |
28 | f1ofo 5367 | . . . . . 6 | |
29 | forn 5343 | . . . . . 6 | |
30 | 13, 20, 28, 29 | 4syl 18 | . . . . 5 |
31 | 30 | raleqdv 2630 | . . . 4 |
32 | 27, 31 | bitr3d 189 | . . 3 |
33 | 19, 32 | sylibd 148 | . 2 Se |
34 | dfse2 4907 | . 2 Se | |
35 | 33, 34 | syl6ibr 161 | 1 Se Se |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2414 cvv 2681 cin 3065 csn 3522 Se wse 4246 ccnv 4533 crn 4535 cima 4537 wfn 5113 wfo 5116 wf1o 5117 cfv 5118 wiso 5119 |
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 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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-se 4250 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-isom 5127 |
This theorem is referenced by: isose 5715 |
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