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Theorem fcof1 5600
 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 500 . . . . . . . 8 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹𝑥) = (𝐹𝑦))
32fveq2d 5344 . . . . . . 7 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑅‘(𝐹𝑥)) = (𝑅‘(𝐹𝑦)))
4 simpll 497 . . . . . . . 8 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐹:𝐴𝐵)
5 simprll 505 . . . . . . . 8 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑥𝐴)
6 fvco3 5410 . . . . . . . 8 ((𝐹:𝐴𝐵𝑥𝐴) → ((𝑅𝐹)‘𝑥) = (𝑅‘(𝐹𝑥)))
74, 5, 6syl2anc 404 . . . . . . 7 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑥) = (𝑅‘(𝐹𝑥)))
8 simprlr 506 . . . . . . . 8 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑦𝐴)
9 fvco3 5410 . . . . . . . 8 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝑅𝐹)‘𝑦) = (𝑅‘(𝐹𝑦)))
104, 8, 9syl2anc 404 . . . . . . 7 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑦) = (𝑅‘(𝐹𝑦)))
113, 7, 103eqtr4d 2137 . . . . . 6 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑥) = ((𝑅𝐹)‘𝑦))
12 simplr 498 . . . . . . 7 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑅𝐹) = ( I ↾ 𝐴))
1312fveq1d 5342 . . . . . 6 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑥) = (( I ↾ 𝐴)‘𝑥))
1412fveq1d 5342 . . . . . 6 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑦) = (( I ↾ 𝐴)‘𝑦))
1511, 13, 143eqtr3d 2135 . . . . 5 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (( I ↾ 𝐴)‘𝑥) = (( I ↾ 𝐴)‘𝑦))
16 fvresi 5529 . . . . . 6 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
175, 16syl 14 . . . . 5 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
18 fvresi 5529 . . . . . 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 2135 . . . 4 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑥 = 𝑦)
2120expr 368 . . 3 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
2221ralrimivva 2467 . 2 ((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) → ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
23 dff13 5585 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
241, 22, 23sylanbrc 409 1 ((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴1-1𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1296   ∈ wcel 1445  ∀wral 2370   I cid 4139   ↾ cres 4469   ∘ ccom 4471  ⟶wf 5045  –1-1→wf1 5046  ‘cfv 5049 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060 This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fv 5057 This theorem is referenced by:  fcof1o  5606
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