Theorem fcof1 7139

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 482	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴⟶𝐵)
2		simprr 769	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹‘𝑥) = (𝐹‘𝑦))
3	2	fveq2d 6760	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑅‘(𝐹‘𝑥)) = (𝑅‘(𝐹‘𝑦)))
4		simpll 763	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝐹:𝐴⟶𝐵)
5		simprll 775	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑥 ∈ 𝐴)
6		fvco3 6849	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝑅 ∘ 𝐹)‘𝑥) = (𝑅‘(𝐹‘𝑥)))
7	4, 5, 6	syl2anc 583	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑅 ∘ 𝐹)‘𝑥) = (𝑅‘(𝐹‘𝑥)))
8		simprlr 776	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑦 ∈ 𝐴)
9		fvco3 6849	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝑅 ∘ 𝐹)‘𝑦) = (𝑅‘(𝐹‘𝑦)))
10	4, 8, 9	syl2anc 583	. . . . . . 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 2788	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑅 ∘ 𝐹)‘𝑥) = ((𝑅 ∘ 𝐹)‘𝑦))
12		simplr 765	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑅 ∘ 𝐹) = ( I ↾ 𝐴))
13	12	fveq1d 6758	. . . . . 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 6758	. . . . . 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 2786	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (( I ↾ 𝐴)‘𝑥) = (( I ↾ 𝐴)‘𝑦))
16		fvresi 7027	. . . . . 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 7027	. . . . . 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 2786	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑥 = 𝑦)
21	20	expr 456	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
22	21	ralrimivva 3114	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
23		dff13 7109	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
24	1, 22, 23	sylanbrc 582	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴–1-1→𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063   I cid 5479   ↾ cres 5582   ∘ ccom 5584  ⟶wf 6414  –1-1→wf1 6415  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fv 6426

This theorem is referenced by:  fcof1od  7146