Theorem fcofo 7229

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 1136	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) → 𝐹:𝐴⟶𝐵)
2		ffvelcdm 7019	. . . . 5 ⊢ ((𝑆:𝐵⟶𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑆‘𝑦) ∈ 𝐴)
3	2	3ad2antl2 1187	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → (𝑆‘𝑦) ∈ 𝐴)
4		simpl3 1194	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → (𝐹 ∘ 𝑆) = ( I ↾ 𝐵))
5	4	fveq1d 6828	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → ((𝐹 ∘ 𝑆)‘𝑦) = (( I ↾ 𝐵)‘𝑦))
6		fvco3 6926	. . . . . 6 ⊢ ((𝑆:𝐵⟶𝐴 ∧ 𝑦 ∈ 𝐵) → ((𝐹 ∘ 𝑆)‘𝑦) = (𝐹‘(𝑆‘𝑦)))
7	6	3ad2antl2 1187	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → ((𝐹 ∘ 𝑆)‘𝑦) = (𝐹‘(𝑆‘𝑦)))
8		fvresi 7113	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦)
9	8	adantl 481	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
10	5, 7, 9	3eqtr3rd 2773	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → 𝑦 = (𝐹‘(𝑆‘𝑦)))
11		fveq2 6826	. . . . 5 ⊢ (𝑥 = (𝑆‘𝑦) → (𝐹‘𝑥) = (𝐹‘(𝑆‘𝑦)))
12	11	rspceeqv 3602	. . . 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 3121	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
15		dffo3 7040	. 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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   I cid 5517   ↾ cres 5625   ∘ ccom 5627  ⟶wf 6482  –onto→wfo 6484  ‘cfv 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fo 6492  df-fv 6494

This theorem is referenced by:  fcof1od  7235  smndex2dnrinv  18807  cofidf2a  49103  cofidf1a  49104  cofidf1  49107