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

Theorem rexrn 6317
Description: Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
Hypothesis
Ref Expression
rexrn.1 (𝑥 = (𝐹𝑦) → (𝜑𝜓))
Assertion
Ref Expression
rexrn (𝐹 Fn 𝐴 → (∃𝑥 ∈ ran 𝐹𝜑 ↔ ∃𝑦𝐴 𝜓))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦   𝜓,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem rexrn
StepHypRef Expression
1 fvex 6158 . . 3 (𝐹𝑦) ∈ V
21a1i 11 . 2 ((𝐹 Fn 𝐴𝑦𝐴) → (𝐹𝑦) ∈ V)
3 fvelrnb 6200 . . 3 (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦𝐴 (𝐹𝑦) = 𝑥))
4 eqcom 2628 . . . 4 ((𝐹𝑦) = 𝑥𝑥 = (𝐹𝑦))
54rexbii 3034 . . 3 (∃𝑦𝐴 (𝐹𝑦) = 𝑥 ↔ ∃𝑦𝐴 𝑥 = (𝐹𝑦))
63, 5syl6bb 276 . 2 (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦𝐴 𝑥 = (𝐹𝑦)))
7 rexrn.1 . . 3 (𝑥 = (𝐹𝑦) → (𝜑𝜓))
87adantl 482 . 2 ((𝐹 Fn 𝐴𝑥 = (𝐹𝑦)) → (𝜑𝜓))
92, 6, 8rexxfr2d 4843 1 (𝐹 Fn 𝐴 → (∃𝑥 ∈ ran 𝐹𝜑 ↔ ∃𝑦𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3186  ran crn 5075   Fn wfn 5842  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-fv 5855
This theorem is referenced by:  elrnrexdm  6319  wemapwe  8538  rexanuz  14019  climsup  14334  supcvg  14513  ruclem12  14895  prmreclem6  15549  vdwmc  15606  znunit  19831  lmbr2  20973  lmff  21015  1stcfb  21158  imasf1oxms  22204  lebnumlem3  22670  lmmbr2  22965  lmcau  23019  bcthlem4  23032  mbfsup  23337  itg2monolem1  23423  itg2gt0  23433  ostth  25228  uhgrvtxedgiedgb  25926  dfnbgr3  26123  vdn0conngrumgrv2  26922  erdszelem10  30890  neibastop2lem  31997  filnetlem4  32018  mblfinlem2  33079  istotbnd3  33202  sstotbnd  33206  heibor  33252  nacsfix  36755  fnwe2lem2  37101  climinf  39242
  Copyright terms: Public domain W3C validator