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Mirrors > Home > ILE Home > Th. List > caofinvl | Unicode version |
Description: Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.) |
Ref | Expression |
---|---|
caofref.1 | |
caofref.2 | |
caofinv.3 | |
caofinv.4 | |
caofinv.5 | |
caofinvl.6 |
Ref | Expression |
---|---|
caofinvl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caofref.1 | . . . 4 | |
2 | caofinv.4 | . . . . . . . . 9 | |
3 | 2 | adantr 274 | . . . . . . . 8 |
4 | caofref.2 | . . . . . . . . 9 | |
5 | 4 | ffvelrnda 5523 | . . . . . . . 8 |
6 | 3, 5 | ffvelrnd 5524 | . . . . . . 7 |
7 | eqid 2117 | . . . . . . 7 | |
8 | 6, 7 | fmptd 5542 | . . . . . 6 |
9 | caofinv.5 | . . . . . . 7 | |
10 | 9 | feq1d 5229 | . . . . . 6 |
11 | 8, 10 | mpbird 166 | . . . . 5 |
12 | 11 | ffvelrnda 5523 | . . . 4 |
13 | 4 | ffvelrnda 5523 | . . . 4 |
14 | 6 | ralrimiva 2482 | . . . . . . 7 |
15 | 7 | fnmpt 5219 | . . . . . . 7 |
16 | 14, 15 | syl 14 | . . . . . 6 |
17 | 9 | fneq1d 5183 | . . . . . 6 |
18 | 16, 17 | mpbird 166 | . . . . 5 |
19 | dffn5im 5435 | . . . . 5 | |
20 | 18, 19 | syl 14 | . . . 4 |
21 | 4 | feqmptd 5442 | . . . 4 |
22 | 1, 12, 13, 20, 21 | offval2 5965 | . . 3 |
23 | 9 | fveq1d 5391 | . . . . . . . 8 |
24 | 23 | adantr 274 | . . . . . . 7 |
25 | simpr 109 | . . . . . . . 8 | |
26 | 2 | adantr 274 | . . . . . . . . 9 |
27 | 26, 13 | ffvelrnd 5524 | . . . . . . . 8 |
28 | fveq2 5389 | . . . . . . . . . 10 | |
29 | 28 | fveq2d 5393 | . . . . . . . . 9 |
30 | 29, 7 | fvmptg 5465 | . . . . . . . 8 |
31 | 25, 27, 30 | syl2anc 408 | . . . . . . 7 |
32 | 24, 31 | eqtrd 2150 | . . . . . 6 |
33 | 32 | oveq1d 5757 | . . . . 5 |
34 | fveq2 5389 | . . . . . . . 8 | |
35 | id 19 | . . . . . . . 8 | |
36 | 34, 35 | oveq12d 5760 | . . . . . . 7 |
37 | 36 | eqeq1d 2126 | . . . . . 6 |
38 | caofinvl.6 | . . . . . . . 8 | |
39 | 38 | ralrimiva 2482 | . . . . . . 7 |
40 | 39 | adantr 274 | . . . . . 6 |
41 | 37, 40, 13 | rspcdva 2768 | . . . . 5 |
42 | 33, 41 | eqtrd 2150 | . . . 4 |
43 | 42 | mpteq2dva 3988 | . . 3 |
44 | 22, 43 | eqtrd 2150 | . 2 |
45 | fconstmpt 4556 | . 2 | |
46 | 44, 45 | syl6eqr 2168 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 wral 2393 csn 3497 cmpt 3959 cxp 4507 wfn 5088 wf 5089 cfv 5093 (class class class)co 5742 cof 5948 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-of 5950 |
This theorem is referenced by: (None) |
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