Theorem fcof1 5692

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:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴–1-1→𝐵)

Proof of Theorem fcof1

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 108	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴⟶𝐵)
2		simprr 522	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹‘𝑥) = (𝐹‘𝑦))
3	2	fveq2d 5433	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑅‘(𝐹‘𝑥)) = (𝑅‘(𝐹‘𝑦)))
4		simpll 519	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝐹:𝐴⟶𝐵)
5		simprll 527	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑥 ∈ 𝐴)
6		fvco3 5500	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝑅 ∘ 𝐹)‘𝑥) = (𝑅‘(𝐹‘𝑥)))
7	4, 5, 6	syl2anc 409	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑅 ∘ 𝐹)‘𝑥) = (𝑅‘(𝐹‘𝑥)))
8		simprlr 528	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑦 ∈ 𝐴)
9		fvco3 5500	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝑅 ∘ 𝐹)‘𝑦) = (𝑅‘(𝐹‘𝑦)))
10	4, 8, 9	syl2anc 409	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑅 ∘ 𝐹)‘𝑦) = (𝑅‘(𝐹‘𝑦)))
11	3, 7, 10	3eqtr4d 2183	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑅 ∘ 𝐹)‘𝑥) = ((𝑅 ∘ 𝐹)‘𝑦))
12		simplr 520	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑅 ∘ 𝐹) = ( I ↾ 𝐴))
13	12	fveq1d 5431	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑅 ∘ 𝐹)‘𝑥) = (( I ↾ 𝐴)‘𝑥))
14	12	fveq1d 5431	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑅 ∘ 𝐹)‘𝑦) = (( I ↾ 𝐴)‘𝑦))
15	11, 13, 14	3eqtr3d 2181	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (( I ↾ 𝐴)‘𝑥) = (( I ↾ 𝐴)‘𝑦))
16		fvresi 5621	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
17	5, 16	syl 14	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
18		fvresi 5621	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦)
19	8, 18	syl 14	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (( I ↾ 𝐴)‘𝑦) = 𝑦)
20	15, 17, 19	3eqtr3d 2181	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑥 = 𝑦)
21	20	expr 373	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
22	21	ralrimivva 2517	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
23		dff13 5677	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
24	1, 22, 23	sylanbrc 414	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴–1-1→𝐵)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 1481  ∀wral 2417   I cid 4218   ↾ cres 4549   ∘ ccom 4551  ⟶wf 5127  –1-1→wf1 5128  ‘cfv 5131

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fv 5139

This theorem is referenced by:  fcof1o  5698