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Theorem foelrn 7108
Description: Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.)
Assertion
Ref Expression
foelrn ((𝐹:𝐴onto𝐵𝐶𝐵) → ∃𝑥𝐴 𝐶 = (𝐹𝑥))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem foelrn
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dffo3 7104 . . 3 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
21simprbi 498 . 2 (𝐹:𝐴onto𝐵 → ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥))
3 eqeq1 2737 . . . 4 (𝑦 = 𝐶 → (𝑦 = (𝐹𝑥) ↔ 𝐶 = (𝐹𝑥)))
43rexbidv 3179 . . 3 (𝑦 = 𝐶 → (∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ ∃𝑥𝐴 𝐶 = (𝐹𝑥)))
54rspccva 3612 . 2 ((∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥) ∧ 𝐶𝐵) → ∃𝑥𝐴 𝐶 = (𝐹𝑥))
62, 5sylan 581 1 ((𝐹:𝐴onto𝐵𝐶𝐵) → ∃𝑥𝐴 𝐶 = (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071  wf 6540  ontowfo 6542  cfv 6544
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 5300  ax-nul 5307  ax-pr 5428
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552
This theorem is referenced by:  foco2  7109  fofinf1o  9327  fodomacn  10051  iunfictbso  10109  cff1  10253  cofsmo  10264  axcclem  10452  konigthlem  10563  tskuni  10778  fulli  17864  efgredlemc  19613  efgrelexlemb  19618  efgredeu  19620  ghmcyg  19764  znfld  21116  znrrg  21121  cygznlem3  21125  ovoliunnul  25024  lgsdchr  26858  foresf1o  31773  iunrdx  31826  znfermltl  32510  crngohomfo  36922  fourierdlem20  44891  fourierdlem52  44922  fourierdlem63  44933  fourierdlem64  44934  fourierdlem65  44935  isomuspgrlem2d  46547
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