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Theorem fcof1o 5372
Description: Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
fcof1o (((𝐹:AB 𝐺:BA) ((𝐹𝐺) = ( I ↾ B) (𝐺𝐹) = ( I ↾ A))) → (𝐹:A1-1-ontoB 𝐹 = 𝐺))

Proof of Theorem fcof1o
StepHypRef Expression
1 fcof1 5366 . . . 4 ((𝐹:AB (𝐺𝐹) = ( I ↾ A)) → 𝐹:A1-1B)
21ad2ant2rl 480 . . 3 (((𝐹:AB 𝐺:BA) ((𝐹𝐺) = ( I ↾ B) (𝐺𝐹) = ( I ↾ A))) → 𝐹:A1-1B)
3 fcofo 5367 . . . . 5 ((𝐹:AB 𝐺:BA (𝐹𝐺) = ( I ↾ B)) → 𝐹:AontoB)
433expa 1103 . . . 4 (((𝐹:AB 𝐺:BA) (𝐹𝐺) = ( I ↾ B)) → 𝐹:AontoB)
54adantrr 448 . . 3 (((𝐹:AB 𝐺:BA) ((𝐹𝐺) = ( I ↾ B) (𝐺𝐹) = ( I ↾ A))) → 𝐹:AontoB)
6 df-f1o 4852 . . 3 (𝐹:A1-1-ontoB ↔ (𝐹:A1-1B 𝐹:AontoB))
72, 5, 6sylanbrc 394 . 2 (((𝐹:AB 𝐺:BA) ((𝐹𝐺) = ( I ↾ B) (𝐺𝐹) = ( I ↾ A))) → 𝐹:A1-1-ontoB)
8 simprl 483 . . . 4 (((𝐹:AB 𝐺:BA) ((𝐹𝐺) = ( I ↾ B) (𝐺𝐹) = ( I ↾ A))) → (𝐹𝐺) = ( I ↾ B))
98coeq2d 4441 . . 3 (((𝐹:AB 𝐺:BA) ((𝐹𝐺) = ( I ↾ B) (𝐺𝐹) = ( I ↾ A))) → (𝐹 ∘ (𝐹𝐺)) = (𝐹 ∘ ( I ↾ B)))
10 coass 4782 . . . 4 ((𝐹𝐹) ∘ 𝐺) = (𝐹 ∘ (𝐹𝐺))
11 f1ococnv1 5098 . . . . . . 7 (𝐹:A1-1-ontoB → (𝐹𝐹) = ( I ↾ A))
127, 11syl 14 . . . . . 6 (((𝐹:AB 𝐺:BA) ((𝐹𝐺) = ( I ↾ B) (𝐺𝐹) = ( I ↾ A))) → (𝐹𝐹) = ( I ↾ A))
1312coeq1d 4440 . . . . 5 (((𝐹:AB 𝐺:BA) ((𝐹𝐺) = ( I ↾ B) (𝐺𝐹) = ( I ↾ A))) → ((𝐹𝐹) ∘ 𝐺) = (( I ↾ A) ∘ 𝐺))
14 fcoi2 5014 . . . . . 6 (𝐺:BA → (( I ↾ A) ∘ 𝐺) = 𝐺)
1514ad2antlr 458 . . . . 5 (((𝐹:AB 𝐺:BA) ((𝐹𝐺) = ( I ↾ B) (𝐺𝐹) = ( I ↾ A))) → (( I ↾ A) ∘ 𝐺) = 𝐺)
1613, 15eqtrd 2069 . . . 4 (((𝐹:AB 𝐺:BA) ((𝐹𝐺) = ( I ↾ B) (𝐺𝐹) = ( I ↾ A))) → ((𝐹𝐹) ∘ 𝐺) = 𝐺)
1710, 16syl5eqr 2083 . . 3 (((𝐹:AB 𝐺:BA) ((𝐹𝐺) = ( I ↾ B) (𝐺𝐹) = ( I ↾ A))) → (𝐹 ∘ (𝐹𝐺)) = 𝐺)
18 f1ocnv 5082 . . . 4 (𝐹:A1-1-ontoB𝐹:B1-1-ontoA)
19 f1of 5069 . . . 4 (𝐹:B1-1-ontoA𝐹:BA)
20 fcoi1 5013 . . . 4 (𝐹:BA → (𝐹 ∘ ( I ↾ B)) = 𝐹)
217, 18, 19, 204syl 18 . . 3 (((𝐹:AB 𝐺:BA) ((𝐹𝐺) = ( I ↾ B) (𝐺𝐹) = ( I ↾ A))) → (𝐹 ∘ ( I ↾ B)) = 𝐹)
229, 17, 213eqtr3rd 2078 . 2 (((𝐹:AB 𝐺:BA) ((𝐹𝐺) = ( I ↾ B) (𝐺𝐹) = ( I ↾ A))) → 𝐹 = 𝐺)
237, 22jca 290 1 (((𝐹:AB 𝐺:BA) ((𝐹𝐺) = ( I ↾ B) (𝐺𝐹) = ( I ↾ A))) → (𝐹:A1-1-ontoB 𝐹 = 𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   I cid 4016  ccnv 4287  cres 4290  ccom 4292  wf 4841  1-1wf1 4842  ontowfo 4843  1-1-ontowf1o 4844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853
This theorem is referenced by: (None)
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