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Mirrors > Home > ILE Home > Th. List > caoftrn | Unicode version |
Description: Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.) |
Ref | Expression |
---|---|
caofref.1 | |
caofref.2 | |
caofcom.3 | |
caofass.4 | |
caoftrn.5 |
Ref | Expression |
---|---|
caoftrn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caoftrn.5 | . . . . . 6 | |
2 | 1 | ralrimivvva 2402 | . . . . 5 |
3 | 2 | adantr 261 | . . . 4 |
4 | caofref.2 | . . . . . 6 | |
5 | 4 | ffvelrnda 5302 | . . . . 5 |
6 | caofcom.3 | . . . . . 6 | |
7 | 6 | ffvelrnda 5302 | . . . . 5 |
8 | caofass.4 | . . . . . 6 | |
9 | 8 | ffvelrnda 5302 | . . . . 5 |
10 | breq1 3767 | . . . . . . . 8 | |
11 | 10 | anbi1d 438 | . . . . . . 7 |
12 | breq1 3767 | . . . . . . 7 | |
13 | 11, 12 | imbi12d 223 | . . . . . 6 |
14 | breq2 3768 | . . . . . . . 8 | |
15 | breq1 3767 | . . . . . . . 8 | |
16 | 14, 15 | anbi12d 442 | . . . . . . 7 |
17 | 16 | imbi1d 220 | . . . . . 6 |
18 | breq2 3768 | . . . . . . . 8 | |
19 | 18 | anbi2d 437 | . . . . . . 7 |
20 | breq2 3768 | . . . . . . 7 | |
21 | 19, 20 | imbi12d 223 | . . . . . 6 |
22 | 13, 17, 21 | rspc3v 2665 | . . . . 5 |
23 | 5, 7, 9, 22 | syl3anc 1135 | . . . 4 |
24 | 3, 23 | mpd 13 | . . 3 |
25 | 24 | ralimdva 2387 | . 2 |
26 | ffn 5046 | . . . . . 6 | |
27 | 4, 26 | syl 14 | . . . . 5 |
28 | ffn 5046 | . . . . . 6 | |
29 | 6, 28 | syl 14 | . . . . 5 |
30 | caofref.1 | . . . . 5 | |
31 | inidm 3146 | . . . . 5 | |
32 | eqidd 2041 | . . . . 5 | |
33 | eqidd 2041 | . . . . 5 | |
34 | 27, 29, 30, 30, 31, 32, 33 | ofrfval 5720 | . . . 4 |
35 | ffn 5046 | . . . . . 6 | |
36 | 8, 35 | syl 14 | . . . . 5 |
37 | eqidd 2041 | . . . . 5 | |
38 | 29, 36, 30, 30, 31, 33, 37 | ofrfval 5720 | . . . 4 |
39 | 34, 38 | anbi12d 442 | . . 3 |
40 | r19.26 2441 | . . 3 | |
41 | 39, 40 | syl6bbr 187 | . 2 |
42 | 27, 36, 30, 30, 31, 32, 37 | ofrfval 5720 | . 2 |
43 | 25, 41, 42 | 3imtr4d 192 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 w3a 885 wceq 1243 wcel 1393 wral 2306 class class class wbr 3764 wfn 4897 wf 4898 cfv 4902 cofr 5711 |
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-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 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 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-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 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-ofr 5713 |
This theorem is referenced by: (None) |
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