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Mirrors > Home > ILE Home > Th. List > sbthlemi8 | Unicode version |
Description: Lemma for isbth 6913. (Contributed by NM, 27-Mar-1998.) |
Ref | Expression |
---|---|
sbthlem.1 | |
sbthlem.2 | |
sbthlem.3 |
Ref | Expression |
---|---|
sbthlemi8 | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funres11 5244 | . . . 4 | |
2 | 1 | ad2antlr 481 | . . 3 EXMID |
3 | funcnvcnv 5231 | . . . . . 6 | |
4 | funres11 5244 | . . . . . 6 | |
5 | 3, 4 | syl 14 | . . . . 5 |
6 | 5 | ad2antrr 480 | . . . 4 |
7 | 6 | ad2antrl 482 | . . 3 EXMID |
8 | simpll 519 | . . . 4 EXMID EXMID | |
9 | simprll 527 | . . . . 5 EXMID | |
10 | 9 | simprd 113 | . . . 4 EXMID |
11 | simprlr 528 | . . . 4 EXMID | |
12 | simprr 522 | . . . 4 EXMID | |
13 | df-ima 4601 | . . . . . . 7 | |
14 | df-rn 4599 | . . . . . . 7 | |
15 | 13, 14 | eqtr2i 2179 | . . . . . 6 |
16 | df-ima 4601 | . . . . . . . 8 | |
17 | df-rn 4599 | . . . . . . . 8 | |
18 | 16, 17 | eqtri 2178 | . . . . . . 7 |
19 | sbthlem.1 | . . . . . . . 8 | |
20 | sbthlem.2 | . . . . . . . 8 | |
21 | 19, 20 | sbthlemi4 6906 | . . . . . . 7 EXMID |
22 | 18, 21 | eqtr3id 2204 | . . . . . 6 EXMID |
23 | ineq12 3304 | . . . . . 6 | |
24 | 15, 22, 23 | sylancr 411 | . . . . 5 EXMID |
25 | disjdif 3467 | . . . . 5 | |
26 | 24, 25 | eqtrdi 2206 | . . . 4 EXMID |
27 | 8, 10, 11, 12, 26 | syl121anc 1225 | . . 3 EXMID |
28 | funun 5216 | . . 3 | |
29 | 2, 7, 27, 28 | syl21anc 1219 | . 2 EXMID |
30 | sbthlem.3 | . . . . 5 | |
31 | 30 | cnveqi 4763 | . . . 4 |
32 | cnvun 4993 | . . . 4 | |
33 | 31, 32 | eqtri 2178 | . . 3 |
34 | 33 | funeqi 5193 | . 2 |
35 | 29, 34 | sylibr 133 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 963 wceq 1335 wcel 2128 cab 2143 cvv 2712 cdif 3099 cun 3100 cin 3101 wss 3102 c0 3395 cuni 3774 EXMIDwem 4157 ccnv 4587 cdm 4588 crn 4589 cres 4590 cima 4591 wfun 5166 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4028 df-exmid 4158 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-fun 5174 |
This theorem is referenced by: sbthlemi9 6911 |
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