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Mirrors > Home > ILE Home > Th. List > sbthlemi9 | Unicode version |
Description: Lemma for isbth 6855. (Contributed by NM, 28-Mar-1998.) |
Ref | Expression |
---|---|
sbthlem.1 | |
sbthlem.2 | |
sbthlem.3 |
Ref | Expression |
---|---|
sbthlemi9 | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 982 | . . . . . . . . . 10 EXMID | |
2 | df-f1 5128 | . . . . . . . . . 10 | |
3 | 1, 2 | sylib 121 | . . . . . . . . 9 EXMID |
4 | 3 | simpld 111 | . . . . . . . 8 EXMID |
5 | df-f 5127 | . . . . . . . 8 | |
6 | 4, 5 | sylib 121 | . . . . . . 7 EXMID |
7 | 6 | simpld 111 | . . . . . 6 EXMID |
8 | df-fn 5126 | . . . . . 6 | |
9 | 7, 8 | sylib 121 | . . . . 5 EXMID |
10 | 9 | simpld 111 | . . . 4 EXMID |
11 | simp3 983 | . . . . . 6 EXMID | |
12 | df-f1 5128 | . . . . . 6 | |
13 | 11, 12 | sylib 121 | . . . . 5 EXMID |
14 | 13 | simprd 113 | . . . 4 EXMID |
15 | sbthlem.1 | . . . . 5 | |
16 | sbthlem.2 | . . . . 5 | |
17 | sbthlem.3 | . . . . 5 | |
18 | 15, 16, 17 | sbthlem7 6851 | . . . 4 |
19 | 10, 14, 18 | syl2anc 408 | . . 3 EXMID |
20 | simp1 981 | . . . 4 EXMID EXMID | |
21 | 9 | simprd 113 | . . . 4 EXMID |
22 | 13 | simpld 111 | . . . . . 6 EXMID |
23 | df-f 5127 | . . . . . 6 | |
24 | 22, 23 | sylib 121 | . . . . 5 EXMID |
25 | 24 | simprd 113 | . . . 4 EXMID |
26 | 15, 16, 17 | sbthlemi5 6849 | . . . 4 EXMID |
27 | 20, 21, 25, 26 | syl12anc 1214 | . . 3 EXMID |
28 | df-fn 5126 | . . 3 | |
29 | 19, 27, 28 | sylanbrc 413 | . 2 EXMID |
30 | 3 | simprd 113 | . . . 4 EXMID |
31 | 24 | simpld 111 | . . . . . 6 EXMID |
32 | df-fn 5126 | . . . . . 6 | |
33 | 31, 32 | sylib 121 | . . . . 5 EXMID |
34 | 33, 25 | jca 304 | . . . 4 EXMID |
35 | 15, 16, 17 | sbthlemi8 6852 | . . . 4 EXMID |
36 | 20, 30, 34, 14, 35 | syl22anc 1217 | . . 3 EXMID |
37 | 6 | simprd 113 | . . . 4 EXMID |
38 | 33 | simprd 113 | . . . . 5 EXMID |
39 | 38, 25 | jca 304 | . . . 4 EXMID |
40 | df-rn 4550 | . . . . 5 | |
41 | 15, 16, 17 | sbthlemi6 6850 | . . . . 5 EXMID |
42 | 40, 41 | syl5eqr 2186 | . . . 4 EXMID |
43 | 20, 37, 39, 14, 42 | syl22anc 1217 | . . 3 EXMID |
44 | df-fn 5126 | . . 3 | |
45 | 36, 43, 44 | sylanbrc 413 | . 2 EXMID |
46 | dff1o4 5375 | . 2 | |
47 | 29, 45, 46 | sylanbrc 413 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 cab 2125 cvv 2686 cdif 3068 cun 3069 wss 3071 cuni 3736 EXMIDwem 4118 ccnv 4538 cdm 4539 crn 4540 cres 4541 cima 4542 wfun 5117 wfn 5118 wf 5119 wf1 5120 wf1o 5122 |
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-in1 603 ax-in2 604 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 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-exmid 4119 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 |
This theorem is referenced by: sbthlemi10 6854 |
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