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Theorem fcof1o 5690
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 5684 . . . 4 ((𝐹:𝐴𝐵 ∧ (𝐺𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴1-1𝐵)
21ad2ant2rl 502 . . 3 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → 𝐹:𝐴1-1𝐵)
3 fcofo 5685 . . . . 5 ((𝐹:𝐴𝐵𝐺:𝐵𝐴 ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹:𝐴onto𝐵)
433expa 1181 . . . 4 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹:𝐴onto𝐵)
54adantrr 470 . . 3 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → 𝐹:𝐴onto𝐵)
6 df-f1o 5130 . . 3 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
72, 5, 6sylanbrc 413 . 2 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → 𝐹:𝐴1-1-onto𝐵)
8 simprl 520 . . . 4 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → (𝐹𝐺) = ( I ↾ 𝐵))
98coeq2d 4701 . . 3 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → (𝐹 ∘ (𝐹𝐺)) = (𝐹 ∘ ( I ↾ 𝐵)))
10 coass 5057 . . . 4 ((𝐹𝐹) ∘ 𝐺) = (𝐹 ∘ (𝐹𝐺))
11 f1ococnv1 5396 . . . . . . 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 4700 . . . . 5 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → ((𝐹𝐹) ∘ 𝐺) = (( I ↾ 𝐴) ∘ 𝐺))
14 fcoi2 5304 . . . . . 6 (𝐺:𝐵𝐴 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
1514ad2antlr 480 . . . . 5 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
1613, 15eqtrd 2172 . . . 4 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → ((𝐹𝐹) ∘ 𝐺) = 𝐺)
1710, 16syl5eqr 2186 . . 3 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → (𝐹 ∘ (𝐹𝐺)) = 𝐺)
18 f1ocnv 5380 . . . 4 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
19 f1of 5367 . . . 4 (𝐹:𝐵1-1-onto𝐴𝐹:𝐵𝐴)
20 fcoi1 5303 . . . 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 2181 . 2 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → 𝐹 = 𝐺)
237, 22jca 304 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 103   = wceq 1331   I cid 4210  ccnv 4538  cres 4541  ccom 4543  wf 5119  1-1wf1 5120  ontowfo 5121  1-1-ontowf1o 5122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131
This theorem is referenced by:  txswaphmeo  12506
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