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Theorem 2fvcoidd 6200
 Description: Show that the composition of two functions is the identity function by applying both functions to each value of the domain of the first function. (Contributed by AV, 15-Dec-2019.)
Hypotheses
Ref Expression
2fvcoidd.f
2fvcoidd.g
2fvcoidd.i
Assertion
Ref Expression
2fvcoidd
Distinct variable groups:   ,   ,   ,

Proof of Theorem 2fvcoidd
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 2fvcoidd.g . . 3
2 2fvcoidd.f . . 3
3 fcompt 6067 . . 3
41, 2, 3syl2anc 661 . 2
5 2fvcoidd.i . . . . . 6
6 fveq2 5871 . . . . . . . . 9
76fveq2d 5875 . . . . . . . 8
8 id 22 . . . . . . . 8
97, 8eqeq12d 2479 . . . . . . 7
109rspccv 3207 . . . . . 6
115, 10syl 16 . . . . 5
1211imp 429 . . . 4
1312mpteq2dva 4538 . . 3
14 mptresid 5333 . . 3
1513, 14syl6eq 2514 . 2
164, 15eqtrd 2498 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  =wceq 1395  e.wcel 1818  A.wral 2807  e.cmpt 4510   cid 4795  |cres 5006  o.ccom 5008  -->wf 5589  cfv 5593 This theorem is referenced by:  2fvidf1od  6201  2fvidinvd  6202 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fv 5601
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