Theorem fcof1 7281

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 483	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴⟶𝐵)
2		simprr 771	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹‘𝑥) = (𝐹‘𝑦))
3	2	fveq2d 6892	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑅‘(𝐹‘𝑥)) = (𝑅‘(𝐹‘𝑦)))
4		simpll 765	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝐹:𝐴⟶𝐵)
5		simprll 777	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑥 ∈ 𝐴)
6		fvco3 6987	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝑅 ∘ 𝐹)‘𝑥) = (𝑅‘(𝐹‘𝑥)))
7	4, 5, 6	syl2anc 584	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑅 ∘ 𝐹)‘𝑥) = (𝑅‘(𝐹‘𝑥)))
8		simprlr 778	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑦 ∈ 𝐴)
9		fvco3 6987	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝑅 ∘ 𝐹)‘𝑦) = (𝑅‘(𝐹‘𝑦)))
10	4, 8, 9	syl2anc 584	. . . . . . 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 2782	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑅 ∘ 𝐹)‘𝑥) = ((𝑅 ∘ 𝐹)‘𝑦))
12		simplr 767	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑅 ∘ 𝐹) = ( I ↾ 𝐴))
13	12	fveq1d 6890	. . . . . 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 6890	. . . . . 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 2780	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (( I ↾ 𝐴)‘𝑥) = (( I ↾ 𝐴)‘𝑦))
16		fvresi 7167	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
17	5, 16	syl 17	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (( I ↾ 𝐴)‘𝑥) = 𝑥)
18		fvresi 7167	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦)
19	8, 18	syl 17	. . . . 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 2780	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑥 = 𝑦)
21	20	expr 457	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
22	21	ralrimivva 3200	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
23		dff13 7250	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
24	1, 22, 23	sylanbrc 583	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴–1-1→𝐵)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061   I cid 5572   ↾ cres 5677   ∘ ccom 5679  ⟶wf 6536  –1-1→wf1 6537  ‘cfv 6540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fv 6548

This theorem is referenced by:  fcof1od  7288