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Theorem fcofo 7235
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 1137 . 2 ((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) → 𝐹:𝐴𝐵)
2 ffvelcdm 7033 . . . . 5 ((𝑆:𝐵𝐴𝑦𝐵) → (𝑆𝑦) ∈ 𝐴)
323ad2antl2 1187 . . . 4 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → (𝑆𝑦) ∈ 𝐴)
4 simpl3 1194 . . . . . 6 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → (𝐹𝑆) = ( I ↾ 𝐵))
54fveq1d 6845 . . . . 5 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → ((𝐹𝑆)‘𝑦) = (( I ↾ 𝐵)‘𝑦))
6 fvco3 6941 . . . . . 6 ((𝑆:𝐵𝐴𝑦𝐵) → ((𝐹𝑆)‘𝑦) = (𝐹‘(𝑆𝑦)))
763ad2antl2 1187 . . . . 5 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → ((𝐹𝑆)‘𝑦) = (𝐹‘(𝑆𝑦)))
8 fvresi 7120 . . . . . 6 (𝑦𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦)
98adantl 483 . . . . 5 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
105, 7, 93eqtr3rd 2782 . . . 4 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → 𝑦 = (𝐹‘(𝑆𝑦)))
11 fveq2 6843 . . . . 5 (𝑥 = (𝑆𝑦) → (𝐹𝑥) = (𝐹‘(𝑆𝑦)))
1211rspceeqv 3596 . . . 4 (((𝑆𝑦) ∈ 𝐴𝑦 = (𝐹‘(𝑆𝑦))) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
133, 10, 12syl2anc 585 . . 3 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
1413ralrimiva 3140 . 2 ((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) → ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥))
15 dffo3 7053 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
161, 14, 15sylanbrc 584 1 ((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) → 𝐹:𝐴onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  wrex 3070   I cid 5531  cres 5636  ccom 5638  wf 6493  ontowfo 6495  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505
This theorem is referenced by:  fcof1od  7241  smndex2dnrinv  18730
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