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Theorem foelrni 6702
 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
foelrni ((𝐹:𝐴onto𝐵𝑌𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑌)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝑌

Proof of Theorem foelrni
StepHypRef Expression
1 forn 6568 . . . 4 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
21eleq2d 2875 . . 3 (𝐹:𝐴onto𝐵 → (𝑌 ∈ ran 𝐹𝑌𝐵))
3 fofn 6567 . . . 4 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
4 fvelrnb 6701 . . . 4 (𝐹 Fn 𝐴 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝑌))
53, 4syl 17 . . 3 (𝐹:𝐴onto𝐵 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝑌))
62, 5bitr3d 284 . 2 (𝐹:𝐴onto𝐵 → (𝑌𝐵 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝑌))
76biimpa 480 1 ((𝐹:𝐴onto𝐵𝑌𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  ran crn 5520   Fn wfn 6319  –onto→wfo 6322  ‘cfv 6324 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fo 6330  df-fv 6332 This theorem is referenced by:  mhmid  18215  mhmmnd  18216  ghmgrp  18218  symgmov2  18511  ghmcmn  18948  founiiun  41818  founiiun0  41832  sge0f1o  43036  isomenndlem  43184  ovnsubaddlem1  43224  f1oresf1o2  43862
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