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Theorem fcof1 5652
Description: An application is injective if a retraction exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 11-Nov-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
Assertion
Ref Expression
fcof1 ((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴1-1𝐵)

Proof of Theorem fcof1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 108 . 2 ((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴𝐵)
2 simprr 506 . . . . . . . 8 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹𝑥) = (𝐹𝑦))
32fveq2d 5393 . . . . . . 7 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑅‘(𝐹𝑥)) = (𝑅‘(𝐹𝑦)))
4 simpll 503 . . . . . . . 8 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐹:𝐴𝐵)
5 simprll 511 . . . . . . . 8 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑥𝐴)
6 fvco3 5460 . . . . . . . 8 ((𝐹:𝐴𝐵𝑥𝐴) → ((𝑅𝐹)‘𝑥) = (𝑅‘(𝐹𝑥)))
74, 5, 6syl2anc 408 . . . . . . 7 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑥) = (𝑅‘(𝐹𝑥)))
8 simprlr 512 . . . . . . . 8 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑦𝐴)
9 fvco3 5460 . . . . . . . 8 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝑅𝐹)‘𝑦) = (𝑅‘(𝐹𝑦)))
104, 8, 9syl2anc 408 . . . . . . 7 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑦) = (𝑅‘(𝐹𝑦)))
113, 7, 103eqtr4d 2160 . . . . . 6 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑥) = ((𝑅𝐹)‘𝑦))
12 simplr 504 . . . . . . 7 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑅𝐹) = ( I ↾ 𝐴))
1312fveq1d 5391 . . . . . 6 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑥) = (( I ↾ 𝐴)‘𝑥))
1412fveq1d 5391 . . . . . 6 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑦) = (( I ↾ 𝐴)‘𝑦))
1511, 13, 143eqtr3d 2158 . . . . 5 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (( I ↾ 𝐴)‘𝑥) = (( I ↾ 𝐴)‘𝑦))
16 fvresi 5581 . . . . . 6 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
175, 16syl 14 . . . . 5 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
18 fvresi 5581 . . . . . 6 (𝑦𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦)
198, 18syl 14 . . . . 5 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (( I ↾ 𝐴)‘𝑦) = 𝑦)
2015, 17, 193eqtr3d 2158 . . . 4 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑥 = 𝑦)
2120expr 372 . . 3 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
2221ralrimivva 2491 . 2 ((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) → ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
23 dff13 5637 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
241, 22, 23sylanbrc 413 1 ((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴1-1𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  wral 2393   I cid 4180  cres 4511  ccom 4513  wf 5089  1-1wf1 5090  cfv 5093
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fv 5101
This theorem is referenced by:  fcof1o  5658
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