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Theorem rexrn 6848
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 fvexd 6681 . 2 ((𝐹 Fn 𝐴𝑦𝐴) → (𝐹𝑦) ∈ V)
2 fvelrnb 6722 . . 3 (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦𝐴 (𝐹𝑦) = 𝑥))
3 eqcom 2832 . . . 4 ((𝐹𝑦) = 𝑥𝑥 = (𝐹𝑦))
43rexbii 3251 . . 3 (∃𝑦𝐴 (𝐹𝑦) = 𝑥 ↔ ∃𝑦𝐴 𝑥 = (𝐹𝑦))
52, 4syl6bb 288 . 2 (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦𝐴 𝑥 = (𝐹𝑦)))
6 rexrn.1 . . 3 (𝑥 = (𝐹𝑦) → (𝜑𝜓))
76adantl 482 . 2 ((𝐹 Fn 𝐴𝑥 = (𝐹𝑦)) → (𝜑𝜓))
81, 5, 7rexxfr2d 5307 1 (𝐹 Fn 𝐴 → (∃𝑥 ∈ ran 𝐹𝜑 ↔ ∃𝑦𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wrex 3143  Vcvv 3499  ran crn 5554   Fn wfn 6346  cfv 6351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-iota 6311  df-fun 6353  df-fn 6354  df-fv 6359
This theorem is referenced by:  elrnrexdm  6850  wemapwe  9152  rexanuz  14698  climsup  15019  supcvg  15203  ruclem12  15586  prmreclem6  16249  vdwmc  16306  znunit  20626  lmbr2  21783  lmff  21825  1stcfb  21969  imasf1oxms  23014  lebnumlem3  23482  lmmbr2  23777  lmcau  23831  bcthlem4  23845  mbfsup  24180  itg2monolem1  24266  itg2gt0  24276  ostth  26129  uhgrvtxedgiedgb  26835  dfnbgr3  27034  vdn0conngrumgrv2  27889  erdszelem10  32331  neibastop2lem  33592  filnetlem4  33613  mblfinlem2  34797  istotbnd3  34917  sstotbnd  34921  heibor  34967  nacsfix  39170  fnwe2lem2  39512  climinf  41748
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