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Theorem fcofo 5685
 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:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) → 𝐹:𝐴onto𝐵)

Proof of Theorem fcofo
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 981 . 2 ((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) → 𝐹:𝐴𝐵)
2 ffvelrn 5553 . . . . 5 ((𝑆:𝐵𝐴𝑦𝐵) → (𝑆𝑦) ∈ 𝐴)
323ad2antl2 1144 . . . 4 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → (𝑆𝑦) ∈ 𝐴)
4 simpl3 986 . . . . . 6 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → (𝐹𝑆) = ( I ↾ 𝐵))
54fveq1d 5423 . . . . 5 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → ((𝐹𝑆)‘𝑦) = (( I ↾ 𝐵)‘𝑦))
6 fvco3 5492 . . . . . 6 ((𝑆:𝐵𝐴𝑦𝐵) → ((𝐹𝑆)‘𝑦) = (𝐹‘(𝑆𝑦)))
763ad2antl2 1144 . . . . 5 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → ((𝐹𝑆)‘𝑦) = (𝐹‘(𝑆𝑦)))
8 fvresi 5613 . . . . . 6 (𝑦𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦)
98adantl 275 . . . . 5 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
105, 7, 93eqtr3rd 2181 . . . 4 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → 𝑦 = (𝐹‘(𝑆𝑦)))
11 fveq2 5421 . . . . . 6 (𝑥 = (𝑆𝑦) → (𝐹𝑥) = (𝐹‘(𝑆𝑦)))
1211eqeq2d 2151 . . . . 5 (𝑥 = (𝑆𝑦) → (𝑦 = (𝐹𝑥) ↔ 𝑦 = (𝐹‘(𝑆𝑦))))
1312rspcev 2789 . . . 4 (((𝑆𝑦) ∈ 𝐴𝑦 = (𝐹‘(𝑆𝑦))) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
143, 10, 13syl2anc 408 . . 3 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
1514ralrimiva 2505 . 2 ((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) → ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥))
16 dffo3 5567 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
171, 15, 16sylanbrc 413 1 ((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) → 𝐹:𝐴onto𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417   I cid 4210   ↾ cres 4541   ∘ ccom 4543  ⟶wf 5119  –onto→wfo 5121  ‘cfv 5123 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fo 5129  df-fv 5131 This theorem is referenced by:  fcof1o  5690
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