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Mirrors > Home > ILE Home > Th. List > isoselem | Unicode version |
Description: Lemma for isose 5783. (Contributed by Mario Carneiro, 23-Jun-2015.) |
Ref | Expression |
---|---|
isofrlem.1 | |
isofrlem.2 |
Ref | Expression |
---|---|
isoselem | Se Se |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfse2 4971 | . . . . . . . . 9 Se | |
2 | 1 | biimpi 119 | . . . . . . . 8 Se |
3 | 2 | r19.21bi 2552 | . . . . . . 7 Se |
4 | 3 | expcom 115 | . . . . . 6 Se |
5 | 4 | adantl 275 | . . . . 5 Se |
6 | imaeq2 4936 | . . . . . . . . . . 11 | |
7 | 6 | eleq1d 2233 | . . . . . . . . . 10 |
8 | 7 | imbi2d 229 | . . . . . . . . 9 |
9 | isofrlem.2 | . . . . . . . . 9 | |
10 | 8, 9 | vtoclg 2781 | . . . . . . . 8 |
11 | 10 | com12 30 | . . . . . . 7 |
12 | 11 | adantr 274 | . . . . . 6 |
13 | isofrlem.1 | . . . . . . . 8 | |
14 | isoini 5780 | . . . . . . . 8 | |
15 | 13, 14 | sylan 281 | . . . . . . 7 |
16 | 15 | eleq1d 2233 | . . . . . 6 |
17 | 12, 16 | sylibd 148 | . . . . 5 |
18 | 5, 17 | syld 45 | . . . 4 Se |
19 | 18 | ralrimdva 2544 | . . 3 Se |
20 | isof1o 5769 | . . . . 5 | |
21 | f1ofn 5427 | . . . . 5 | |
22 | sneq 3581 | . . . . . . . . 9 | |
23 | 22 | imaeq2d 4940 | . . . . . . . 8 |
24 | 23 | ineq2d 3318 | . . . . . . 7 |
25 | 24 | eleq1d 2233 | . . . . . 6 |
26 | 25 | ralrn 5617 | . . . . 5 |
27 | 13, 20, 21, 26 | 4syl 18 | . . . 4 |
28 | f1ofo 5433 | . . . . . 6 | |
29 | forn 5407 | . . . . . 6 | |
30 | 13, 20, 28, 29 | 4syl 18 | . . . . 5 |
31 | 30 | raleqdv 2665 | . . . 4 |
32 | 27, 31 | bitr3d 189 | . . 3 |
33 | 19, 32 | sylibd 148 | . 2 Se |
34 | dfse2 4971 | . 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 1342 wcel 2135 wral 2442 cvv 2721 cin 3110 csn 3570 Se wse 4301 ccnv 4597 crn 4599 cima 4601 wfn 5177 wfo 5180 wf1o 5181 cfv 5182 wiso 5183 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-se 4305 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 |
This theorem is referenced by: isose 5783 |
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