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Theorem foelcdmi 6905
Description: A member of a surjective function's codomain is a value of the function. (Contributed by Thierry Arnoux, 23-Jan-2020.)
Assertion
Ref Expression
foelcdmi ((𝐹:𝐴onto𝐵𝑌𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑌)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝑌

Proof of Theorem foelcdmi
StepHypRef Expression
1 forn 6760 . . . 4 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
21eleq2d 2820 . . 3 (𝐹:𝐴onto𝐵 → (𝑌 ∈ ran 𝐹𝑌𝐵))
3 fofn 6759 . . . 4 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
4 fvelrnb 6904 . . . 4 (𝐹 Fn 𝐴 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝑌))
53, 4syl 17 . . 3 (𝐹:𝐴onto𝐵 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝑌))
62, 5bitr3d 281 . 2 (𝐹:𝐴onto𝐵 → (𝑌𝐵 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝑌))
76biimpa 478 1 ((𝐹:𝐴onto𝐵𝑌𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wrex 3070  ran crn 5635   Fn wfn 6492  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-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505
This theorem is referenced by:  mhmid  18873  mhmmnd  18874  ghmgrp  18876  symgmov2  19174  ghmcmn  19615  founiiun  43484  founiiun0  43497  sge0f1o  44709  isomenndlem  44857  ovnsubaddlem1  44897  f1oresf1o2  45609
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