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Theorem fcof1o 5789
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 5783 . . . 4 ((𝐹:𝐴𝐵 ∧ (𝐺𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴1-1𝐵)
21ad2ant2rl 511 . . 3 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → 𝐹:𝐴1-1𝐵)
3 fcofo 5784 . . . . 5 ((𝐹:𝐴𝐵𝐺:𝐵𝐴 ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹:𝐴onto𝐵)
433expa 1203 . . . 4 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹:𝐴onto𝐵)
54adantrr 479 . . 3 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → 𝐹:𝐴onto𝐵)
6 df-f1o 5223 . . 3 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
72, 5, 6sylanbrc 417 . 2 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → 𝐹:𝐴1-1-onto𝐵)
8 simprl 529 . . . 4 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → (𝐹𝐺) = ( I ↾ 𝐵))
98coeq2d 4789 . . 3 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → (𝐹 ∘ (𝐹𝐺)) = (𝐹 ∘ ( I ↾ 𝐵)))
10 coass 5147 . . . 4 ((𝐹𝐹) ∘ 𝐺) = (𝐹 ∘ (𝐹𝐺))
11 f1ococnv1 5490 . . . . . . 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 4788 . . . . 5 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → ((𝐹𝐹) ∘ 𝐺) = (( I ↾ 𝐴) ∘ 𝐺))
14 fcoi2 5397 . . . . . 6 (𝐺:𝐵𝐴 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
1514ad2antlr 489 . . . . 5 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
1613, 15eqtrd 2210 . . . 4 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → ((𝐹𝐹) ∘ 𝐺) = 𝐺)
1710, 16eqtr3id 2224 . . 3 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → (𝐹 ∘ (𝐹𝐺)) = 𝐺)
18 f1ocnv 5474 . . . 4 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
19 f1of 5461 . . . 4 (𝐹:𝐵1-1-onto𝐴𝐹:𝐵𝐴)
20 fcoi1 5396 . . . 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 2219 . 2 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → 𝐹 = 𝐺)
237, 22jca 306 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 104   = wceq 1353   I cid 4288  ccnv 4625  cres 4628  ccom 4630  wf 5212  1-1wf1 5213  ontowfo 5214  1-1-ontowf1o 5215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224
This theorem is referenced by:  txswaphmeo  13791
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