Theorem fcofo 5925

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 1023	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) → 𝐹:𝐴⟶𝐵)
2		ffvelcdm 5780	. . . . 5 ⊢ ((𝑆:𝐵⟶𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑆‘𝑦) ∈ 𝐴)
3	2	3ad2antl2 1186	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → (𝑆‘𝑦) ∈ 𝐴)
4		simpl3 1028	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → (𝐹 ∘ 𝑆) = ( I ↾ 𝐵))
5	4	fveq1d 5641	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → ((𝐹 ∘ 𝑆)‘𝑦) = (( I ↾ 𝐵)‘𝑦))
6		fvco3 5717	. . . . . 6 ⊢ ((𝑆:𝐵⟶𝐴 ∧ 𝑦 ∈ 𝐵) → ((𝐹 ∘ 𝑆)‘𝑦) = (𝐹‘(𝑆‘𝑦)))
7	6	3ad2antl2 1186	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → ((𝐹 ∘ 𝑆)‘𝑦) = (𝐹‘(𝑆‘𝑦)))
8		fvresi 5847	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦)
9	8	adantl 277	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
10	5, 7, 9	3eqtr3rd 2273	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → 𝑦 = (𝐹‘(𝑆‘𝑦)))
11		fveq2 5639	. . . . . 6 ⊢ (𝑥 = (𝑆‘𝑦) → (𝐹‘𝑥) = (𝐹‘(𝑆‘𝑦)))
12	11	eqeq2d 2243	. . . . 5 ⊢ (𝑥 = (𝑆‘𝑦) → (𝑦 = (𝐹‘𝑥) ↔ 𝑦 = (𝐹‘(𝑆‘𝑦))))
13	12	rspcev 2910	. . . 4 ⊢ (((𝑆‘𝑦) ∈ 𝐴 ∧ 𝑦 = (𝐹‘(𝑆‘𝑦))) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
14	3, 10, 13	syl2anc 411	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
15	14	ralrimiva 2605	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
16		dffo3 5794	. 2 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
17	1, 15, 16	sylanbrc 417	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) → 𝐹:𝐴–onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511   I cid 4385   ↾ cres 4727   ∘ ccom 4729  ⟶wf 5322  –onto→wfo 5324  ‘cfv 5326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334

This theorem is referenced by:  fcof1o  5930