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Theorem foelrn 6493
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 6489 . . 3 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
21simprbi 483 . 2 (𝐹:𝐴onto𝐵 → ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥))
3 eqeq1 2728 . . . 4 (𝑦 = 𝐶 → (𝑦 = (𝐹𝑥) ↔ 𝐶 = (𝐹𝑥)))
43rexbidv 3154 . . 3 (𝑦 = 𝐶 → (∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ ∃𝑥𝐴 𝐶 = (𝐹𝑥)))
54rspccva 3412 . 2 ((∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥) ∧ 𝐶𝐵) → ∃𝑥𝐴 𝐶 = (𝐹𝑥))
62, 5sylan 489 1 ((𝐹:𝐴onto𝐵𝐶𝐵) → ∃𝑥𝐴 𝐶 = (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wcel 2103  wral 3014  wrex 3015  wf 5997  ontowfo 5999  cfv 6001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-fo 6007  df-fv 6009
This theorem is referenced by:  foco2  6494  foco2OLD  6495  fofinf1o  8357  fodomacn  8992  iunfictbso  9050  cff1  9193  cofsmo  9204  axcclem  9392  konigthlem  9503  tskuni  9718  fulli  16695  efgredlemc  18279  efgrelexlemb  18284  efgredeu  18286  ghmcyg  18418  znfld  20032  znrrg  20037  cygznlem3  20041  ovoliunnul  23396  lgsdchr  25200  foresf1o  29571  iunrdx  29610  crngohomfo  34037  fourierdlem20  40764  fourierdlem52  40795  fourierdlem63  40806  fourierdlem64  40807  fourierdlem65  40808
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