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Theorem fcof1 7234
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 484 . 2 ((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴𝐵)
2 simprr 772 . . . . . . . 8 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹𝑥) = (𝐹𝑦))
32fveq2d 6847 . . . . . . 7 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑅‘(𝐹𝑥)) = (𝑅‘(𝐹𝑦)))
4 simpll 766 . . . . . . . 8 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐹:𝐴𝐵)
5 simprll 778 . . . . . . . 8 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑥𝐴)
6 fvco3 6941 . . . . . . . 8 ((𝐹:𝐴𝐵𝑥𝐴) → ((𝑅𝐹)‘𝑥) = (𝑅‘(𝐹𝑥)))
74, 5, 6syl2anc 585 . . . . . . 7 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑥) = (𝑅‘(𝐹𝑥)))
8 simprlr 779 . . . . . . . 8 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑦𝐴)
9 fvco3 6941 . . . . . . . 8 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝑅𝐹)‘𝑦) = (𝑅‘(𝐹𝑦)))
104, 8, 9syl2anc 585 . . . . . . 7 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑦) = (𝑅‘(𝐹𝑦)))
113, 7, 103eqtr4d 2783 . . . . . 6 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑥) = ((𝑅𝐹)‘𝑦))
12 simplr 768 . . . . . . 7 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑅𝐹) = ( I ↾ 𝐴))
1312fveq1d 6845 . . . . . 6 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑥) = (( I ↾ 𝐴)‘𝑥))
1412fveq1d 6845 . . . . . 6 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑦) = (( I ↾ 𝐴)‘𝑦))
1511, 13, 143eqtr3d 2781 . . . . 5 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (( I ↾ 𝐴)‘𝑥) = (( I ↾ 𝐴)‘𝑦))
16 fvresi 7120 . . . . . 6 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
175, 16syl 17 . . . . 5 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
18 fvresi 7120 . . . . . 6 (𝑦𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦)
198, 18syl 17 . . . . 5 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (( I ↾ 𝐴)‘𝑦) = 𝑦)
2015, 17, 193eqtr3d 2781 . . . 4 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑥 = 𝑦)
2120expr 458 . . 3 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
2221ralrimivva 3194 . 2 ((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) → ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
23 dff13 7203 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
241, 22, 23sylanbrc 584 1 ((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴1-1𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3061   I cid 5531  cres 5636  ccom 5638  wf 6493  1-1wf1 6494  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fv 6505
This theorem is referenced by:  fcof1od  7241
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