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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 2547 | . . . . 5 |
3 | 2 | adantr 274 | . . . 4 |
4 | caofref.2 | . . . . . 6 | |
5 | 4 | ffvelrnda 5614 | . . . . 5 |
6 | caofcom.3 | . . . . . 6 | |
7 | 6 | ffvelrnda 5614 | . . . . 5 |
8 | caofass.4 | . . . . . 6 | |
9 | 8 | ffvelrnda 5614 | . . . . 5 |
10 | breq1 3979 | . . . . . . . 8 | |
11 | 10 | anbi1d 461 | . . . . . . 7 |
12 | breq1 3979 | . . . . . . 7 | |
13 | 11, 12 | imbi12d 233 | . . . . . 6 |
14 | breq2 3980 | . . . . . . . 8 | |
15 | breq1 3979 | . . . . . . . 8 | |
16 | 14, 15 | anbi12d 465 | . . . . . . 7 |
17 | 16 | imbi1d 230 | . . . . . 6 |
18 | breq2 3980 | . . . . . . . 8 | |
19 | 18 | anbi2d 460 | . . . . . . 7 |
20 | breq2 3980 | . . . . . . 7 | |
21 | 19, 20 | imbi12d 233 | . . . . . 6 |
22 | 13, 17, 21 | rspc3v 2841 | . . . . 5 |
23 | 5, 7, 9, 22 | syl3anc 1227 | . . . 4 |
24 | 3, 23 | mpd 13 | . . 3 |
25 | 24 | ralimdva 2531 | . 2 |
26 | ffn 5331 | . . . . . 6 | |
27 | 4, 26 | syl 14 | . . . . 5 |
28 | ffn 5331 | . . . . . 6 | |
29 | 6, 28 | syl 14 | . . . . 5 |
30 | caofref.1 | . . . . 5 | |
31 | inidm 3326 | . . . . 5 | |
32 | eqidd 2165 | . . . . 5 | |
33 | eqidd 2165 | . . . . 5 | |
34 | 27, 29, 30, 30, 31, 32, 33 | ofrfval 6052 | . . . 4 |
35 | ffn 5331 | . . . . . 6 | |
36 | 8, 35 | syl 14 | . . . . 5 |
37 | eqidd 2165 | . . . . 5 | |
38 | 29, 36, 30, 30, 31, 33, 37 | ofrfval 6052 | . . . 4 |
39 | 34, 38 | anbi12d 465 | . . 3 |
40 | r19.26 2590 | . . 3 | |
41 | 39, 40 | bitr4di 197 | . 2 |
42 | 27, 36, 30, 30, 31, 32, 37 | ofrfval 6052 | . 2 |
43 | 25, 41, 42 | 3imtr4d 202 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 967 wceq 1342 wcel 2135 wral 2442 class class class wbr 3976 wfn 5177 wf 5178 cfv 5182 cofr 6043 |
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-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ofr 6045 |
This theorem is referenced by: (None) |
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