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Theorem fcofo 7036
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 1133 . 2 ((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) → 𝐹:𝐴𝐵)
2 ffvelrn 6840 . . . . 5 ((𝑆:𝐵𝐴𝑦𝐵) → (𝑆𝑦) ∈ 𝐴)
323ad2antl2 1183 . . . 4 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → (𝑆𝑦) ∈ 𝐴)
4 simpl3 1190 . . . . . 6 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → (𝐹𝑆) = ( I ↾ 𝐵))
54fveq1d 6663 . . . . 5 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → ((𝐹𝑆)‘𝑦) = (( I ↾ 𝐵)‘𝑦))
6 fvco3 6751 . . . . . 6 ((𝑆:𝐵𝐴𝑦𝐵) → ((𝐹𝑆)‘𝑦) = (𝐹‘(𝑆𝑦)))
763ad2antl2 1183 . . . . 5 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → ((𝐹𝑆)‘𝑦) = (𝐹‘(𝑆𝑦)))
8 fvresi 6926 . . . . . 6 (𝑦𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦)
98adantl 485 . . . . 5 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
105, 7, 93eqtr3rd 2868 . . . 4 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → 𝑦 = (𝐹‘(𝑆𝑦)))
11 fveq2 6661 . . . . 5 (𝑥 = (𝑆𝑦) → (𝐹𝑥) = (𝐹‘(𝑆𝑦)))
1211rspceeqv 3624 . . . 4 (((𝑆𝑦) ∈ 𝐴𝑦 = (𝐹‘(𝑆𝑦))) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
133, 10, 12syl2anc 587 . . 3 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
1413ralrimiva 3177 . 2 ((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) → ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥))
15 dffo3 6859 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
161, 14, 15sylanbrc 586 1 ((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) → 𝐹:𝐴onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  wrex 3134   I cid 5446  cres 5544  ccom 5546  wf 6339  ontowfo 6341  cfv 6343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fo 6349  df-fv 6351
This theorem is referenced by:  fcof1od  7042  smndex2dnrinv  18080
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