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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 5610 | . . . 4 | |
4 | 1, 2, 3 | syl2anc 408 | . . 3 |
5 | offval.2 | . . . 4 | |
6 | offval.4 | . . . 4 | |
7 | fnex 5610 | . . . 4 | |
8 | 5, 6, 7 | syl2anc 408 | . . 3 |
9 | dmeq 4709 | . . . . . 6 | |
10 | dmeq 4709 | . . . . . 6 | |
11 | 9, 10 | ineqan12d 3249 | . . . . 5 |
12 | fveq1 5388 | . . . . . 6 | |
13 | fveq1 5388 | . . . . . 6 | |
14 | 12, 13 | breqan12d 3915 | . . . . 5 |
15 | 11, 14 | raleqbidv 2615 | . . . 4 |
16 | df-ofr 5951 | . . . 4 | |
17 | 15, 16 | brabga 4156 | . . 3 |
18 | 4, 8, 17 | syl2anc 408 | . 2 |
19 | fndm 5192 | . . . . . 6 | |
20 | 1, 19 | syl 14 | . . . . 5 |
21 | fndm 5192 | . . . . . 6 | |
22 | 5, 21 | syl 14 | . . . . 5 |
23 | 20, 22 | ineq12d 3248 | . . . 4 |
24 | offval.5 | . . . 4 | |
25 | 23, 24 | syl6eq 2166 | . . 3 |
26 | 25 | raleqdv 2609 | . 2 |
27 | inss1 3266 | . . . . . . 7 | |
28 | 24, 27 | eqsstrri 3100 | . . . . . 6 |
29 | 28 | sseli 3063 | . . . . 5 |
30 | offval.6 | . . . . 5 | |
31 | 29, 30 | sylan2 284 | . . . 4 |
32 | inss2 3267 | . . . . . . 7 | |
33 | 24, 32 | eqsstrri 3100 | . . . . . 6 |
34 | 33 | sseli 3063 | . . . . 5 |
35 | offval.7 | . . . . 5 | |
36 | 34, 35 | sylan2 284 | . . . 4 |
37 | 31, 36 | breq12d 3912 | . . 3 |
38 | 37 | ralbidva 2410 | . 2 |
39 | 18, 26, 38 | 3bitrd 213 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 wral 2393 cvv 2660 cin 3040 class class class wbr 3899 cdm 4509 wfn 5088 cfv 5093 cofr 5949 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ofr 5951 |
This theorem is referenced by: ofrval 5960 ofrfval2 5966 caofref 5971 caofrss 5974 caoftrn 5975 |
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