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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 6823 . . . 4 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
21eleq2d 2827 . . 3 (𝐹:𝐴onto𝐵 → (𝑌 ∈ ran 𝐹𝑌𝐵))
3 fofn 6822 . . . 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 1540  wcel 2108  wrex 3070  ran crn 5686   Fn wfn 6556  ontowfo 6559  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569
This theorem is referenced by:  mhmid  19081  mhmmnd  19082  ghmgrp  19084  symgmov2  19405  ghmcmn  19849  imasabl  19894  mndlactfo  33032  mndractfo  33034  founiiun  45184  founiiun0  45195  sge0f1o  46397  isomenndlem  46545  ovnsubaddlem1  46585  f1oresf1o2  47303  grimuhgr  47878  grimcnv  47879
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