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Theorem ofresid 27938
 Description: Applying an operation restricted to the range of the functions does not change the function operation. (Contributed by Thierry Arnoux, 14-Feb-2018.)
Hypotheses
Ref Expression
ofresid.1
ofresid.2
ofresid.3
Assertion
Ref Expression
ofresid

Proof of Theorem ofresid
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ofresid.1 . . . . . . . 8
21ffvelrnda 6011 . . . . . . 7
3 ofresid.2 . . . . . . . 8
43ffvelrnda 6011 . . . . . . 7
5 opelxp 4855 . . . . . . 7
62, 4, 5sylanbrc 664 . . . . . 6
7 fvres 5865 . . . . . 6
86, 7syl 17 . . . . 5
98eqcomd 2412 . . . 4
10 df-ov 6283 . . . 4
11 df-ov 6283 . . . 4
129, 10, 113eqtr4g 2470 . . 3
1312mpteq2dva 4483 . 2
14 ffn 5716 . . . 4
151, 14syl 17 . . 3
16 ffn 5716 . . . 4
173, 16syl 17 . . 3
18 ofresid.3 . . 3
19 inidm 3650 . . 3
20 eqidd 2405 . . 3
21 eqidd 2405 . . 3
2215, 17, 18, 18, 19, 20, 21offval 6530 . 2
2315, 17, 18, 18, 19, 20, 21offval 6530 . 2
2413, 22, 233eqtr4d 2455 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1407   wcel 1844  cop 3980   cmpt 4455   cxp 4823   cres 4827   wfn 5566  wf 5567  cfv 5571  (class class class)co 6280   cof 6521 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pr 4632 This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-of 6523 This theorem is referenced by: (None)
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