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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 261 | . . . . . . . 8 |
4 | caofref.2 | . . . . . . . . 9 | |
5 | 4 | ffvelrnda 5302 | . . . . . . . 8 |
6 | 3, 5 | ffvelrnd 5303 | . . . . . . 7 |
7 | eqid 2040 | . . . . . . 7 | |
8 | 6, 7 | fmptd 5322 | . . . . . 6 |
9 | caofinv.5 | . . . . . . 7 | |
10 | 9 | feq1d 5034 | . . . . . 6 |
11 | 8, 10 | mpbird 156 | . . . . 5 |
12 | 11 | ffvelrnda 5302 | . . . 4 |
13 | 4 | ffvelrnda 5302 | . . . 4 |
14 | 6 | ralrimiva 2392 | . . . . . . 7 |
15 | 7 | fnmpt 5025 | . . . . . . 7 |
16 | 14, 15 | syl 14 | . . . . . 6 |
17 | 9 | fneq1d 4989 | . . . . . 6 |
18 | 16, 17 | mpbird 156 | . . . . 5 |
19 | dffn5im 5219 | . . . . 5 | |
20 | 18, 19 | syl 14 | . . . 4 |
21 | 4 | feqmptd 5226 | . . . 4 |
22 | 1, 12, 13, 20, 21 | offval2 5726 | . . 3 |
23 | 9 | fveq1d 5180 | . . . . . . . 8 |
24 | 23 | adantr 261 | . . . . . . 7 |
25 | simpr 103 | . . . . . . . 8 | |
26 | 2 | adantr 261 | . . . . . . . . 9 |
27 | 26, 13 | ffvelrnd 5303 | . . . . . . . 8 |
28 | fveq2 5178 | . . . . . . . . . 10 | |
29 | 28 | fveq2d 5182 | . . . . . . . . 9 |
30 | 29, 7 | fvmptg 5248 | . . . . . . . 8 |
31 | 25, 27, 30 | syl2anc 391 | . . . . . . 7 |
32 | 24, 31 | eqtrd 2072 | . . . . . 6 |
33 | 32 | oveq1d 5527 | . . . . 5 |
34 | caofinvl.6 | . . . . . . . 8 | |
35 | 34 | ralrimiva 2392 | . . . . . . 7 |
36 | 35 | adantr 261 | . . . . . 6 |
37 | fveq2 5178 | . . . . . . . . 9 | |
38 | id 19 | . . . . . . . . 9 | |
39 | 37, 38 | oveq12d 5530 | . . . . . . . 8 |
40 | 39 | eqeq1d 2048 | . . . . . . 7 |
41 | 40 | rspcva 2654 | . . . . . 6 |
42 | 13, 36, 41 | syl2anc 391 | . . . . 5 |
43 | 33, 42 | eqtrd 2072 | . . . 4 |
44 | 43 | mpteq2dva 3847 | . . 3 |
45 | 22, 44 | eqtrd 2072 | . 2 |
46 | fconstmpt 4387 | . 2 | |
47 | 45, 46 | syl6eqr 2090 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 wral 2306 csn 3375 cmpt 3818 cxp 4343 wfn 4897 wf 4898 cfv 4902 (class class class)co 5512 cof 5710 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-of 5712 |
This theorem is referenced by: (None) |
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