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Theorem foelrn 6286
 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 6282 . . 3 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
21simprbi 479 . 2 (𝐹:𝐴onto𝐵 → ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥))
3 eqeq1 2614 . . . 4 (𝑦 = 𝐶 → (𝑦 = (𝐹𝑥) ↔ 𝐶 = (𝐹𝑥)))
43rexbidv 3034 . . 3 (𝑦 = 𝐶 → (∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ ∃𝑥𝐴 𝐶 = (𝐹𝑥)))
54rspccva 3281 . 2 ((∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥) ∧ 𝐶𝐵) → ∃𝑥𝐴 𝐶 = (𝐹𝑥))
62, 5sylan 487 1 ((𝐹:𝐴onto𝐵𝐶𝐵) → ∃𝑥𝐴 𝐶 = (𝐹𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812 This theorem is referenced by:  foco2  6287  foco2OLD  6288  fofinf1o  8126  fodomacn  8762  iunfictbso  8820  cff1  8963  cofsmo  8974  axcclem  9162  konigthlem  9269  tskuni  9484  fulli  16396  efgredlemc  17981  efgrelexlemb  17986  efgredeu  17988  ghmcyg  18120  znfld  19728  znrrg  19733  cygznlem3  19737  ovoliunnul  23082  lgsdchr  24880  foresf1o  28727  iunrdx  28764  crngohomfo  32975  fourierdlem20  39020  fourierdlem52  39051  fourierdlem63  39062  fourierdlem64  39063  fourierdlem65  39064
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