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Theorem fcof1o 6199
Description: Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.) (Proof shortened by AV, 15-Dec-2019.)
Assertion
Ref Expression
fcof1o

Proof of Theorem fcof1o
StepHypRef Expression
1 simpll 753 . . 3
2 simplr 755 . . 3
3 simprr 757 . . 3
4 simprl 756 . . 3
51, 2, 3, 4fcof1od 6197 . 2
61, 2, 3, 42fcoidinvd 6198 . 2
75, 6jca 532 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  =wceq 1395   cid 4795  `'ccnv 5003  |`cres 5006  o.ccom 5008  -->wf 5589  -1-1-onto->wf1o 5592
This theorem is referenced by:  setcinv  15417  yonedainv  15550  txswaphmeo  20306  rngcinv  32789  rngcinvOLD  32801  ringcinv  32840  ringcinvOLD  32864
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-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
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