Theorem fcofo 5834

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 999	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) → 𝐹:𝐴⟶𝐵)
2		ffvelcdm 5698	. . . . 5 ⊢ ((𝑆:𝐵⟶𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑆‘𝑦) ∈ 𝐴)
3	2	3ad2antl2 1162	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → (𝑆‘𝑦) ∈ 𝐴)
4		simpl3 1004	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → (𝐹 ∘ 𝑆) = ( I ↾ 𝐵))
5	4	fveq1d 5563	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → ((𝐹 ∘ 𝑆)‘𝑦) = (( I ↾ 𝐵)‘𝑦))
6		fvco3 5635	. . . . . 6 ⊢ ((𝑆:𝐵⟶𝐴 ∧ 𝑦 ∈ 𝐵) → ((𝐹 ∘ 𝑆)‘𝑦) = (𝐹‘(𝑆‘𝑦)))
7	6	3ad2antl2 1162	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → ((𝐹 ∘ 𝑆)‘𝑦) = (𝐹‘(𝑆‘𝑦)))
8		fvresi 5758	. . . . . 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 2238	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → 𝑦 = (𝐹‘(𝑆‘𝑦)))
11		fveq2 5561	. . . . . 6 ⊢ (𝑥 = (𝑆‘𝑦) → (𝐹‘𝑥) = (𝐹‘(𝑆‘𝑦)))
12	11	eqeq2d 2208	. . . . 5 ⊢ (𝑥 = (𝑆‘𝑦) → (𝑦 = (𝐹‘𝑥) ↔ 𝑦 = (𝐹‘(𝑆‘𝑦))))
13	12	rspcev 2868	. . . 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 2570	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
16		dffo3 5712	. 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 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476   I cid 4324   ↾ cres 4666   ∘ ccom 4668  ⟶wf 5255  –onto→wfo 5257  ‘cfv 5259

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fo 5265  df-fv 5267

This theorem is referenced by:  fcof1o  5839