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Theorem fcof1 6420
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 471 . 2 ((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴𝐵)
2 simprr 791 . . . . . . . 8 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹𝑥) = (𝐹𝑦))
32fveq2d 6092 . . . . . . 7 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑅‘(𝐹𝑥)) = (𝑅‘(𝐹𝑦)))
4 simpll 785 . . . . . . . 8 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐹:𝐴𝐵)
5 simprll 797 . . . . . . . 8 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑥𝐴)
6 fvco3 6170 . . . . . . . 8 ((𝐹:𝐴𝐵𝑥𝐴) → ((𝑅𝐹)‘𝑥) = (𝑅‘(𝐹𝑥)))
74, 5, 6syl2anc 690 . . . . . . 7 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑥) = (𝑅‘(𝐹𝑥)))
8 simprlr 798 . . . . . . . 8 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑦𝐴)
9 fvco3 6170 . . . . . . . 8 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝑅𝐹)‘𝑦) = (𝑅‘(𝐹𝑦)))
104, 8, 9syl2anc 690 . . . . . . 7 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑦) = (𝑅‘(𝐹𝑦)))
113, 7, 103eqtr4d 2653 . . . . . 6 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑥) = ((𝑅𝐹)‘𝑦))
12 simplr 787 . . . . . . 7 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑅𝐹) = ( I ↾ 𝐴))
1312fveq1d 6090 . . . . . 6 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑥) = (( I ↾ 𝐴)‘𝑥))
1412fveq1d 6090 . . . . . 6 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑅𝐹)‘𝑦) = (( I ↾ 𝐴)‘𝑦))
1511, 13, 143eqtr3d 2651 . . . . 5 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (( I ↾ 𝐴)‘𝑥) = (( I ↾ 𝐴)‘𝑦))
16 fvresi 6322 . . . . . 6 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
175, 16syl 17 . . . . 5 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
18 fvresi 6322 . . . . . 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 2651 . . . 4 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑥 = 𝑦)
2120expr 640 . . 3 (((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
2221ralrimivva 2953 . 2 ((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) → ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
23 dff13 6394 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
241, 22, 23sylanbrc 694 1 ((𝐹:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴1-1𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wral 2895   I cid 4938  cres 5030  ccom 5032  wf 5786  1-1wf1 5787  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fv 5798
This theorem is referenced by:  fcof1od  6427
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