Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  2fvidinvd Unicode version

Theorem 2fvidinvd 6202
 Description: Show that two functions are inverse to each other by applying them twice to each value of their domains. (Contributed by AV, 13-Dec-2019.)
Hypotheses
Ref Expression
2fvcoidd.f
2fvcoidd.g
2fvcoidd.i
2fvidf1od.i
Assertion
Ref Expression
2fvidinvd
Distinct variable groups:   ,   ,   ,   ,   ,   ,

Proof of Theorem 2fvidinvd
StepHypRef Expression
1 2fvcoidd.f . 2
2 2fvcoidd.g . 2
3 2fvcoidd.i . . 3
41, 2, 32fvcoidd 6200 . 2
5 2fvidf1od.i . . 3
62, 1, 52fvcoidd 6200 . 2
71, 2, 4, 62fcoidinvd 6198 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  =wceq 1395  A.wral 2807  'ccnv 5003  -->wf 5589  cfv 5593 This theorem is referenced by:  m2cpminv  19261 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-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601
 Copyright terms: Public domain W3C validator