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Theorem fcof1o 7289
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 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → (𝐹:𝐴1-1-onto𝐵𝐹 = 𝐺))

Proof of Theorem fcof1o
StepHypRef Expression
1 simpll 764 . . 3 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → 𝐹:𝐴𝐵)
2 simplr 766 . . 3 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → 𝐺:𝐵𝐴)
3 simprr 770 . . 3 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → (𝐺𝐹) = ( I ↾ 𝐴))
4 simprl 768 . . 3 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → (𝐹𝐺) = ( I ↾ 𝐵))
51, 2, 3, 4fcof1od 7287 . 2 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → 𝐹:𝐴1-1-onto𝐵)
61, 2, 3, 42fcoidinvd 7288 . 2 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → 𝐹 = 𝐺)
75, 6jca 511 1 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → (𝐹:𝐴1-1-onto𝐵𝐹 = 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533   I cid 5566  ccnv 5668  cres 5671  ccom 5673  wf 6532  1-1-ontowf1o 6535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544
This theorem is referenced by:  setcinv  18050  yonedainv  18244  rngcinv  20531  ringcinv  20565  txswaphmeo  23660  rngcinvALTV  47207  ringcinvALTV  47241
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