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Theorem fcof1 7288
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 482 . 2 ((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴𝐵)
2 simprr 770 . . . . . . . 8 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹𝑥) = (𝐹𝑦))
32fveq2d 6895 . . . . . . 7 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑅‘(𝐹𝑥)) = (𝑅‘(𝐹𝑦)))
4 simpll 764 . . . . . . . 8 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐹:𝐴𝐵)
5 simprll 776 . . . . . . . 8 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑥𝐴)
6 fvco3 6990 . . . . . . . 8 ((𝐹:𝐴𝐵𝑥𝐴) → ((𝑅𝐹)‘𝑥) = (𝑅‘(𝐹𝑥)))
74, 5, 6syl2anc 583 . . . . . . 7 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑥) = (𝑅‘(𝐹𝑥)))
8 simprlr 777 . . . . . . . 8 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑦𝐴)
9 fvco3 6990 . . . . . . . 8 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝑅𝐹)‘𝑦) = (𝑅‘(𝐹𝑦)))
104, 8, 9syl2anc 583 . . . . . . 7 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑦) = (𝑅‘(𝐹𝑦)))
113, 7, 103eqtr4d 2781 . . . . . 6 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑥) = ((𝑅𝐹)‘𝑦))
12 simplr 766 . . . . . . 7 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑅𝐹) = ( I ↾ 𝐴))
1312fveq1d 6893 . . . . . 6 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑥) = (( I ↾ 𝐴)‘𝑥))
1412fveq1d 6893 . . . . . 6 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑦) = (( I ↾ 𝐴)‘𝑦))
1511, 13, 143eqtr3d 2779 . . . . 5 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (( I ↾ 𝐴)‘𝑥) = (( I ↾ 𝐴)‘𝑦))
16 fvresi 7173 . . . . . 6 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
175, 16syl 17 . . . . 5 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
18 fvresi 7173 . . . . . 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 2779 . . . 4 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑥 = 𝑦)
2120expr 456 . . 3 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
2221ralrimivva 3199 . 2 ((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) → ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
23 dff13 7257 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
241, 22, 23sylanbrc 582 1 ((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴1-1𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  wral 3060   I cid 5573  cres 5678  ccom 5680  wf 6539  1-1wf1 6540  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fv 6551
This theorem is referenced by:  fcof1od  7295  psdmplcl  22013
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