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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 5635 | . . . 4 | |
4 | 1, 2, 3 | syl2anc 408 | . . 3 |
5 | offval.2 | . . . 4 | |
6 | offval.4 | . . . 4 | |
7 | fnex 5635 | . . . 4 | |
8 | 5, 6, 7 | syl2anc 408 | . . 3 |
9 | dmeq 4734 | . . . . . 6 | |
10 | dmeq 4734 | . . . . . 6 | |
11 | 9, 10 | ineqan12d 3274 | . . . . 5 |
12 | fveq1 5413 | . . . . . 6 | |
13 | fveq1 5413 | . . . . . 6 | |
14 | 12, 13 | breqan12d 3940 | . . . . 5 |
15 | 11, 14 | raleqbidv 2636 | . . . 4 |
16 | df-ofr 5976 | . . . 4 | |
17 | 15, 16 | brabga 4181 | . . 3 |
18 | 4, 8, 17 | syl2anc 408 | . 2 |
19 | fndm 5217 | . . . . . 6 | |
20 | 1, 19 | syl 14 | . . . . 5 |
21 | fndm 5217 | . . . . . 6 | |
22 | 5, 21 | syl 14 | . . . . 5 |
23 | 20, 22 | ineq12d 3273 | . . . 4 |
24 | offval.5 | . . . 4 | |
25 | 23, 24 | syl6eq 2186 | . . 3 |
26 | 25 | raleqdv 2630 | . 2 |
27 | inss1 3291 | . . . . . . 7 | |
28 | 24, 27 | eqsstrri 3125 | . . . . . 6 |
29 | 28 | sseli 3088 | . . . . 5 |
30 | offval.6 | . . . . 5 | |
31 | 29, 30 | sylan2 284 | . . . 4 |
32 | inss2 3292 | . . . . . . 7 | |
33 | 24, 32 | eqsstrri 3125 | . . . . . 6 |
34 | 33 | sseli 3088 | . . . . 5 |
35 | offval.7 | . . . . 5 | |
36 | 34, 35 | sylan2 284 | . . . 4 |
37 | 31, 36 | breq12d 3937 | . . 3 |
38 | 37 | ralbidva 2431 | . 2 |
39 | 18, 26, 38 | 3bitrd 213 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2414 cvv 2681 cin 3065 class class class wbr 3924 cdm 4534 wfn 5113 cfv 5118 cofr 5974 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ofr 5976 |
This theorem is referenced by: ofrval 5985 ofrfval2 5991 caofref 5996 caofrss 5999 caoftrn 6000 |
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