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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 5404 | . . . 4 | |
4 | 1, 2, 3 | syl2anc 403 | . . 3 |
5 | offval.2 | . . . 4 | |
6 | offval.4 | . . . 4 | |
7 | fnex 5404 | . . . 4 | |
8 | 5, 6, 7 | syl2anc 403 | . . 3 |
9 | dmeq 4553 | . . . . . 6 | |
10 | dmeq 4553 | . . . . . 6 | |
11 | 9, 10 | ineqan12d 3169 | . . . . 5 |
12 | fveq1 5197 | . . . . . 6 | |
13 | fveq1 5197 | . . . . . 6 | |
14 | 12, 13 | breqan12d 3800 | . . . . 5 |
15 | 11, 14 | raleqbidv 2561 | . . . 4 |
16 | df-ofr 5733 | . . . 4 | |
17 | 15, 16 | brabga 4019 | . . 3 |
18 | 4, 8, 17 | syl2anc 403 | . 2 |
19 | fndm 5018 | . . . . . 6 | |
20 | 1, 19 | syl 14 | . . . . 5 |
21 | fndm 5018 | . . . . . 6 | |
22 | 5, 21 | syl 14 | . . . . 5 |
23 | 20, 22 | ineq12d 3168 | . . . 4 |
24 | offval.5 | . . . 4 | |
25 | 23, 24 | syl6eq 2129 | . . 3 |
26 | 25 | raleqdv 2555 | . 2 |
27 | inss1 3186 | . . . . . . 7 | |
28 | 24, 27 | eqsstr3i 3030 | . . . . . 6 |
29 | 28 | sseli 2995 | . . . . 5 |
30 | offval.6 | . . . . 5 | |
31 | 29, 30 | sylan2 280 | . . . 4 |
32 | inss2 3187 | . . . . . . 7 | |
33 | 24, 32 | eqsstr3i 3030 | . . . . . 6 |
34 | 33 | sseli 2995 | . . . . 5 |
35 | offval.7 | . . . . 5 | |
36 | 34, 35 | sylan2 280 | . . . 4 |
37 | 31, 36 | breq12d 3798 | . . 3 |
38 | 37 | ralbidva 2364 | . 2 |
39 | 18, 26, 38 | 3bitrd 212 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 wral 2348 cvv 2601 cin 2972 class class class wbr 3785 cdm 4363 wfn 4917 cfv 4922 cofr 5731 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ofr 5733 |
This theorem is referenced by: ofrval 5742 ofrfval2 5747 caofref 5752 caofrss 5755 caoftrn 5756 |
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