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Mirrors > Home > ILE Home > Th. List > ofrfval | Unicode version |
Description: Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-Jul-2014.) |
Ref | Expression |
---|---|
offval.1 | |
offval.2 | |
offval.3 | |
offval.4 | |
offval.5 | |
offval.6 | |
offval.7 |
Ref | Expression |
---|---|
ofrfval |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | offval.1 | . . . 4 | |
2 | offval.3 | . . . 4 | |
3 | fnex 5383 | . . . 4 | |
4 | 1, 2, 3 | syl2anc 391 | . . 3 |
5 | offval.2 | . . . 4 | |
6 | offval.4 | . . . 4 | |
7 | fnex 5383 | . . . 4 | |
8 | 5, 6, 7 | syl2anc 391 | . . 3 |
9 | dmeq 4535 | . . . . . 6 | |
10 | dmeq 4535 | . . . . . 6 | |
11 | 9, 10 | ineqan12d 3140 | . . . . 5 |
12 | fveq1 5177 | . . . . . 6 | |
13 | fveq1 5177 | . . . . . 6 | |
14 | 12, 13 | breqan12d 3779 | . . . . 5 |
15 | 11, 14 | raleqbidv 2517 | . . . 4 |
16 | df-ofr 5713 | . . . 4 | |
17 | 15, 16 | brabga 4001 | . . 3 |
18 | 4, 8, 17 | syl2anc 391 | . 2 |
19 | fndm 4998 | . . . . . 6 | |
20 | 1, 19 | syl 14 | . . . . 5 |
21 | fndm 4998 | . . . . . 6 | |
22 | 5, 21 | syl 14 | . . . . 5 |
23 | 20, 22 | ineq12d 3139 | . . . 4 |
24 | offval.5 | . . . 4 | |
25 | 23, 24 | syl6eq 2088 | . . 3 |
26 | 25 | raleqdv 2511 | . 2 |
27 | inss1 3157 | . . . . . . 7 | |
28 | 24, 27 | eqsstr3i 2976 | . . . . . 6 |
29 | 28 | sseli 2941 | . . . . 5 |
30 | offval.6 | . . . . 5 | |
31 | 29, 30 | sylan2 270 | . . . 4 |
32 | inss2 3158 | . . . . . . 7 | |
33 | 24, 32 | eqsstr3i 2976 | . . . . . 6 |
34 | 33 | sseli 2941 | . . . . 5 |
35 | offval.7 | . . . . 5 | |
36 | 34, 35 | sylan2 270 | . . . 4 |
37 | 31, 36 | breq12d 3777 | . . 3 |
38 | 37 | ralbidva 2322 | . 2 |
39 | 18, 26, 38 | 3bitrd 203 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wral 2306 cvv 2557 cin 2916 class class class wbr 3764 cdm 4345 wfn 4897 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: ofrval 5722 ofrfval2 5727 caofref 5732 caofrss 5735 caoftrn 5736 |
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