Theorem fcofo 7160

Description: An application is surjective if a section exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 17-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Assertion

Ref	Expression
fcofo	⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) → 𝐹:𝐴–onto→𝐵)

Proof of Theorem fcofo

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

Step	Hyp	Ref	Expression
1		simp1 1135	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) → 𝐹:𝐴⟶𝐵)
2		ffvelrn 6959	. . . . 5 ⊢ ((𝑆:𝐵⟶𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑆‘𝑦) ∈ 𝐴)
3	2	3ad2antl2 1185	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → (𝑆‘𝑦) ∈ 𝐴)
4		simpl3 1192	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → (𝐹 ∘ 𝑆) = ( I ↾ 𝐵))
5	4	fveq1d 6776	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → ((𝐹 ∘ 𝑆)‘𝑦) = (( I ↾ 𝐵)‘𝑦))
6		fvco3 6867	. . . . . 6 ⊢ ((𝑆:𝐵⟶𝐴 ∧ 𝑦 ∈ 𝐵) → ((𝐹 ∘ 𝑆)‘𝑦) = (𝐹‘(𝑆‘𝑦)))
7	6	3ad2antl2 1185	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → ((𝐹 ∘ 𝑆)‘𝑦) = (𝐹‘(𝑆‘𝑦)))
8		fvresi 7045	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦)
9	8	adantl 482	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
10	5, 7, 9	3eqtr3rd 2787	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → 𝑦 = (𝐹‘(𝑆‘𝑦)))
11		fveq2 6774	. . . . 5 ⊢ (𝑥 = (𝑆‘𝑦) → (𝐹‘𝑥) = (𝐹‘(𝑆‘𝑦)))
12	11	rspceeqv 3575	. . . 4 ⊢ (((𝑆‘𝑦) ∈ 𝐴 ∧ 𝑦 = (𝐹‘(𝑆‘𝑦))) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
13	3, 10, 12	syl2anc 584	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
14	13	ralrimiva 3103	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
15		dffo3 6978	. 2 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
16	1, 14, 15	sylanbrc 583	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) → 𝐹:𝐴–onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   I cid 5488   ↾ cres 5591   ∘ ccom 5593  ⟶wf 6429  –onto→wfo 6431  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441

This theorem is referenced by:  fcof1od  7166  smndex2dnrinv  18554