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Theorem rexrnmpt 6623
Description: A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)
Hypotheses
Ref Expression
ralrnmpt.1 𝐹 = (𝑥𝐴𝐵)
ralrnmpt.2 (𝑦 = 𝐵 → (𝜓𝜒))
Assertion
Ref Expression
rexrnmpt (∀𝑥𝐴 𝐵𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥𝐴 𝜒))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐵   𝜒,𝑦   𝑦,𝐹   𝜓,𝑥
Allowed substitution hints:   𝜓(𝑦)   𝜒(𝑥)   𝐴(𝑦)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem rexrnmpt
StepHypRef Expression
1 ralrnmpt.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
2 ralrnmpt.2 . . . . 5 (𝑦 = 𝐵 → (𝜓𝜒))
32notbid 310 . . . 4 (𝑦 = 𝐵 → (¬ 𝜓 ↔ ¬ 𝜒))
41, 3ralrnmpt 6622 . . 3 (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹 ¬ 𝜓 ↔ ∀𝑥𝐴 ¬ 𝜒))
54notbid 310 . 2 (∀𝑥𝐴 𝐵𝑉 → (¬ ∀𝑦 ∈ ran 𝐹 ¬ 𝜓 ↔ ¬ ∀𝑥𝐴 ¬ 𝜒))
6 dfrex2 3204 . 2 (∃𝑦 ∈ ran 𝐹𝜓 ↔ ¬ ∀𝑦 ∈ ran 𝐹 ¬ 𝜓)
7 dfrex2 3204 . 2 (∃𝑥𝐴 𝜒 ↔ ¬ ∀𝑥𝐴 ¬ 𝜒)
85, 6, 73bitr4g 306 1 (∀𝑥𝐴 𝐵𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198   = wceq 1656  wcel 2164  wral 3117  wrex 3118  cmpt 4954  ran crn 5347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-fv 6135
This theorem is referenced by:  onoviun  7711  onnseq  7712  ghmcyg  18657  pgpfac1lem2  18835  pgpfac1lem3  18837  pgpfac1lem4  18838  pptbas  21190  lly1stc  21677  txbas  21748  eltsms  22313  tsmsf1o  22325  psmetutop  22749  xrge0tsms  23014  fmcfil  23447  ellimc2  24047  limcflf  24051  xrge0tsmsd  30326  poimirlem23  33975  poimirlem24  33976  poimirlem30  33982  cntotbnd  34136
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