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Mirrors > Home > ILE Home > Th. List > supisolem | Unicode version |
Description: Lemma for supisoti 6423. (Contributed by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
supiso.1 | |
supiso.2 |
Ref | Expression |
---|---|
supisolem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supiso.1 | . . 3 | |
2 | supiso.2 | . . 3 | |
3 | 1, 2 | jca 300 | . 2 |
4 | simpll 495 | . . . . . . . 8 | |
5 | 4 | adantr 270 | . . . . . . 7 |
6 | simplr 496 | . . . . . . 7 | |
7 | simplr 496 | . . . . . . . 8 | |
8 | 7 | sselda 2999 | . . . . . . 7 |
9 | isorel 5468 | . . . . . . 7 | |
10 | 5, 6, 8, 9 | syl12anc 1167 | . . . . . 6 |
11 | 10 | notbid 624 | . . . . 5 |
12 | 11 | ralbidva 2364 | . . . 4 |
13 | isof1o 5467 | . . . . . . 7 | |
14 | 4, 13 | syl 14 | . . . . . 6 |
15 | f1ofn 5147 | . . . . . 6 | |
16 | 14, 15 | syl 14 | . . . . 5 |
17 | breq2 3789 | . . . . . . 7 | |
18 | 17 | notbid 624 | . . . . . 6 |
19 | 18 | ralima 5416 | . . . . 5 |
20 | 16, 7, 19 | syl2anc 403 | . . . 4 |
21 | 12, 20 | bitr4d 189 | . . 3 |
22 | 4 | adantr 270 | . . . . . . 7 |
23 | simpr 108 | . . . . . . 7 | |
24 | simplr 496 | . . . . . . 7 | |
25 | isorel 5468 | . . . . . . 7 | |
26 | 22, 23, 24, 25 | syl12anc 1167 | . . . . . 6 |
27 | 22 | adantr 270 | . . . . . . . . 9 |
28 | simplr 496 | . . . . . . . . 9 | |
29 | 7 | adantr 270 | . . . . . . . . . 10 |
30 | 29 | sselda 2999 | . . . . . . . . 9 |
31 | isorel 5468 | . . . . . . . . 9 | |
32 | 27, 28, 30, 31 | syl12anc 1167 | . . . . . . . 8 |
33 | 32 | rexbidva 2365 | . . . . . . 7 |
34 | 16 | adantr 270 | . . . . . . . 8 |
35 | breq2 3789 | . . . . . . . . 9 | |
36 | 35 | rexima 5415 | . . . . . . . 8 |
37 | 34, 29, 36 | syl2anc 403 | . . . . . . 7 |
38 | 33, 37 | bitr4d 189 | . . . . . 6 |
39 | 26, 38 | imbi12d 232 | . . . . 5 |
40 | 39 | ralbidva 2364 | . . . 4 |
41 | f1ofo 5153 | . . . . 5 | |
42 | breq1 3788 | . . . . . . 7 | |
43 | breq1 3788 | . . . . . . . 8 | |
44 | 43 | rexbidv 2369 | . . . . . . 7 |
45 | 42, 44 | imbi12d 232 | . . . . . 6 |
46 | 45 | cbvfo 5445 | . . . . 5 |
47 | 14, 41, 46 | 3syl 17 | . . . 4 |
48 | 40, 47 | bitrd 186 | . . 3 |
49 | 21, 48 | anbi12d 456 | . 2 |
50 | 3, 49 | sylan 277 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wceq 1284 wcel 1433 wral 2348 wrex 2349 wss 2973 class class class wbr 3785 cima 4366 wfn 4917 wfo 4920 wf1o 4921 cfv 4922 wiso 4923 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-isom 4931 |
This theorem is referenced by: supisoex 6422 supisoti 6423 |
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