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Theorem foelcdmi 6970
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 6824 . . . 4 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
21eleq2d 2825 . . 3 (𝐹:𝐴onto𝐵 → (𝑌 ∈ ran 𝐹𝑌𝐵))
3 fofn 6823 . . . 4 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
4 fvelrnb 6969 . . . 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 476 1 ((𝐹:𝐴onto𝐵𝑌𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  ran crn 5690   Fn wfn 6558  ontowfo 6561  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571
This theorem is referenced by:  mhmid  19094  mhmmnd  19095  ghmgrp  19097  symgmov2  19420  ghmcmn  19864  imasabl  19909  mndlactfo  33015  mndractfo  33017  founiiun  45122  founiiun0  45133  sge0f1o  46338  isomenndlem  46486  ovnsubaddlem1  46526  f1oresf1o2  47241  grimuhgr  47816  grimcnv  47817
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