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Theorem rexrn 6476
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 6316 . 2 ((𝐹 Fn 𝐴𝑦𝐴) → (𝐹𝑦) ∈ V)
2 fvelrnb 6357 . . 3 (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦𝐴 (𝐹𝑦) = 𝑥))
3 eqcom 2731 . . . 4 ((𝐹𝑦) = 𝑥𝑥 = (𝐹𝑦))
43rexbii 3143 . . 3 (∃𝑦𝐴 (𝐹𝑦) = 𝑥 ↔ ∃𝑦𝐴 𝑥 = (𝐹𝑦))
52, 4syl6bb 276 . 2 (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦𝐴 𝑥 = (𝐹𝑦)))
6 rexrn.1 . . 3 (𝑥 = (𝐹𝑦) → (𝜑𝜓))
76adantl 473 . 2 ((𝐹 Fn 𝐴𝑥 = (𝐹𝑦)) → (𝜑𝜓))
81, 5, 7rexxfr2d 4988 1 (𝐹 Fn 𝐴 → (∃𝑥 ∈ ran 𝐹𝜑 ↔ ∃𝑦𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1596  wcel 2103  wrex 3015  Vcvv 3304  ran crn 5219   Fn wfn 5996  cfv 6001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-iota 5964  df-fun 6003  df-fn 6004  df-fv 6009
This theorem is referenced by:  elrnrexdm  6478  wemapwe  8707  rexanuz  14205  climsup  14520  supcvg  14708  ruclem12  15090  prmreclem6  15748  vdwmc  15805  znunit  20035  lmbr2  21186  lmff  21228  1stcfb  21371  imasf1oxms  22416  lebnumlem3  22884  lmmbr2  23178  lmcau  23232  bcthlem4  23245  mbfsup  23551  itg2monolem1  23637  itg2gt0  23647  ostth  25448  uhgrvtxedgiedgb  26151  dfnbgr3  26351  vdn0conngrumgrv2  27269  erdszelem10  31410  neibastop2lem  32582  filnetlem4  32603  mblfinlem2  33679  istotbnd3  33802  sstotbnd  33806  heibor  33852  nacsfix  37694  fnwe2lem2  38040  climinf  40258
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