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Theorem fcof1o 5456
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:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → (𝐹:𝐴1-1-onto𝐵𝐹 = 𝐺))

Proof of Theorem fcof1o
StepHypRef Expression
1 fcof1 5450 . . . 4 ((𝐹:𝐴𝐵 ∧ (𝐺𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴1-1𝐵)
21ad2ant2rl 488 . . 3 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → 𝐹:𝐴1-1𝐵)
3 fcofo 5451 . . . . 5 ((𝐹:𝐴𝐵𝐺:𝐵𝐴 ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹:𝐴onto𝐵)
433expa 1115 . . . 4 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹:𝐴onto𝐵)
54adantrr 456 . . 3 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → 𝐹:𝐴onto𝐵)
6 df-f1o 4936 . . 3 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
72, 5, 6sylanbrc 402 . 2 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → 𝐹:𝐴1-1-onto𝐵)
8 simprl 491 . . . 4 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → (𝐹𝐺) = ( I ↾ 𝐵))
98coeq2d 4525 . . 3 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → (𝐹 ∘ (𝐹𝐺)) = (𝐹 ∘ ( I ↾ 𝐵)))
10 coass 4866 . . . 4 ((𝐹𝐹) ∘ 𝐺) = (𝐹 ∘ (𝐹𝐺))
11 f1ococnv1 5182 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))
127, 11syl 14 . . . . . 6 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → (𝐹𝐹) = ( I ↾ 𝐴))
1312coeq1d 4524 . . . . 5 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → ((𝐹𝐹) ∘ 𝐺) = (( I ↾ 𝐴) ∘ 𝐺))
14 fcoi2 5098 . . . . . 6 (𝐺:𝐵𝐴 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
1514ad2antlr 466 . . . . 5 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
1613, 15eqtrd 2088 . . . 4 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → ((𝐹𝐹) ∘ 𝐺) = 𝐺)
1710, 16syl5eqr 2102 . . 3 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → (𝐹 ∘ (𝐹𝐺)) = 𝐺)
18 f1ocnv 5166 . . . 4 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
19 f1of 5153 . . . 4 (𝐹:𝐵1-1-onto𝐴𝐹:𝐵𝐴)
20 fcoi1 5097 . . . 4 (𝐹:𝐵𝐴 → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
217, 18, 19, 204syl 18 . . 3 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
229, 17, 213eqtr3rd 2097 . 2 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → 𝐹 = 𝐺)
237, 22jca 294 1 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → (𝐹:𝐴1-1-onto𝐵𝐹 = 𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259   I cid 4052  ccnv 4371  cres 4374  ccom 4376  wf 4925  1-1wf1 4926  ontowfo 4927  1-1-ontowf1o 4928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937
This theorem is referenced by: (None)
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