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Theorem fcofo 7308
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 7101 . . . . 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 6908 . . . . 5 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → ((𝐹𝑆)‘𝑦) = (( I ↾ 𝐵)‘𝑦))
6 fvco3 7008 . . . . . 6 ((𝑆:𝐵𝐴𝑦𝐵) → ((𝐹𝑆)‘𝑦) = (𝐹‘(𝑆𝑦)))
763ad2antl2 1187 . . . . 5 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → ((𝐹𝑆)‘𝑦) = (𝐹‘(𝑆𝑦)))
8 fvresi 7193 . . . . . 6 (𝑦𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦)
98adantl 481 . . . . 5 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
105, 7, 93eqtr3rd 2786 . . . 4 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → 𝑦 = (𝐹‘(𝑆𝑦)))
11 fveq2 6906 . . . . 5 (𝑥 = (𝑆𝑦) → (𝐹𝑥) = (𝐹‘(𝑆𝑦)))
1211rspceeqv 3645 . . . 4 (((𝑆𝑦) ∈ 𝐴𝑦 = (𝐹‘(𝑆𝑦))) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
133, 10, 12syl2anc 584 . . 3 (((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) ∧ 𝑦𝐵) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
1413ralrimiva 3146 . 2 ((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) → ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥))
15 dffo3 7122 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
161, 14, 15sylanbrc 583 1 ((𝐹:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) → 𝐹:𝐴onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070   I cid 5577  cres 5687  ccom 5689  wf 6557  ontowfo 6559  cfv 6561
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569
This theorem is referenced by:  fcof1od  7314  smndex2dnrinv  18928
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