MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  foelrni Structured version   Visualization version   GIF version

Theorem foelrni 6201
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 6075 . . . 4 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
21eleq2d 2684 . . 3 (𝐹:𝐴onto𝐵 → (𝑌 ∈ ran 𝐹𝑌𝐵))
3 fofn 6074 . . . 4 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
4 fvelrnb 6200 . . . 4 (𝐹 Fn 𝐴 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝑌))
53, 4syl 17 . . 3 (𝐹:𝐴onto𝐵 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝑌))
62, 5bitr3d 270 . 2 (𝐹:𝐴onto𝐵 → (𝑌𝐵 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝑌))
76biimpa 501 1 ((𝐹:𝐴onto𝐵𝑌𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  ran crn 5075   Fn wfn 5842  ontowfo 5845  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fo 5853  df-fv 5855
This theorem is referenced by:  mhmid  17457  mhmmnd  17458  ghmgrp  17460  symgmov2  17734  ghmcmn  18158  founiiun  38831  founiiun0  38848  sge0f1o  39903  isomenndlem  40048  ovnsubaddlem1  40088
  Copyright terms: Public domain W3C validator