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Mirrors > Home > ILE Home > Th. List > ofco | Unicode version |
Description: The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.) |
Ref | Expression |
---|---|
ofco.1 | |
ofco.2 | |
ofco.3 | |
ofco.4 | |
ofco.5 | |
ofco.6 | |
ofco.7 |
Ref | Expression |
---|---|
ofco |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ofco.3 | . . . 4 | |
2 | 1 | ffvelrnda 5548 | . . 3 |
3 | 1 | feqmptd 5467 | . . 3 |
4 | ofco.1 | . . . 4 | |
5 | ofco.2 | . . . 4 | |
6 | ofco.4 | . . . 4 | |
7 | ofco.5 | . . . 4 | |
8 | ofco.7 | . . . 4 | |
9 | eqidd 2138 | . . . 4 | |
10 | eqidd 2138 | . . . 4 | |
11 | 4, 5, 6, 7, 8, 9, 10 | offval 5982 | . . 3 |
12 | fveq2 5414 | . . . 4 | |
13 | fveq2 5414 | . . . 4 | |
14 | 12, 13 | oveq12d 5785 | . . 3 |
15 | 2, 3, 11, 14 | fmptco 5579 | . 2 |
16 | inss1 3291 | . . . . . 6 | |
17 | 8, 16 | eqsstrri 3125 | . . . . 5 |
18 | fss 5279 | . . . . 5 | |
19 | 1, 17, 18 | sylancl 409 | . . . 4 |
20 | fnfco 5292 | . . . 4 | |
21 | 4, 19, 20 | syl2anc 408 | . . 3 |
22 | inss2 3292 | . . . . . 6 | |
23 | 8, 22 | eqsstrri 3125 | . . . . 5 |
24 | fss 5279 | . . . . 5 | |
25 | 1, 23, 24 | sylancl 409 | . . . 4 |
26 | fnfco 5292 | . . . 4 | |
27 | 5, 25, 26 | syl2anc 408 | . . 3 |
28 | ofco.6 | . . 3 | |
29 | inidm 3280 | . . 3 | |
30 | ffn 5267 | . . . . 5 | |
31 | 1, 30 | syl 14 | . . . 4 |
32 | fvco2 5483 | . . . 4 | |
33 | 31, 32 | sylan 281 | . . 3 |
34 | fvco2 5483 | . . . 4 | |
35 | 31, 34 | sylan 281 | . . 3 |
36 | 21, 27, 28, 28, 29, 33, 35 | offval 5982 | . 2 |
37 | 15, 36 | eqtr4d 2173 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cin 3065 wss 3066 cmpt 3984 ccom 4538 wfn 5113 wf 5114 cfv 5118 (class class class)co 5767 cof 5973 |
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-in1 603 ax-in2 604 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 ax-setind 4447 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 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-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 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-ov 5770 df-oprab 5771 df-mpo 5772 df-of 5975 |
This theorem is referenced by: (None) |
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