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Theorem rexrnmpt 6335
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 308 . . . 4 (𝑦 = 𝐵 → (¬ 𝜓 ↔ ¬ 𝜒))
41, 3ralrnmpt 6334 . . 3 (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹 ¬ 𝜓 ↔ ∀𝑥𝐴 ¬ 𝜒))
54notbid 308 . 2 (∀𝑥𝐴 𝐵𝑉 → (¬ ∀𝑦 ∈ ran 𝐹 ¬ 𝜓 ↔ ¬ ∀𝑥𝐴 ¬ 𝜒))
6 dfrex2 2992 . 2 (∃𝑦 ∈ ran 𝐹𝜓 ↔ ¬ ∀𝑦 ∈ ran 𝐹 ¬ 𝜓)
7 dfrex2 2992 . 2 (∃𝑥𝐴 𝜒 ↔ ¬ ∀𝑥𝐴 ¬ 𝜒)
85, 6, 73bitr4g 303 1 (∀𝑥𝐴 𝐵𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1480  wcel 1987  wral 2908  wrex 2909  cmpt 4683  ran crn 5085
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-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
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 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-fv 5865
This theorem is referenced by:  onoviun  7400  onnseq  7401  ghmcyg  18237  pgpfac1lem2  18414  pgpfac1lem3  18416  pgpfac1lem4  18417  pptbas  20752  lly1stc  21239  txbas  21310  eltsms  21876  tsmsf1o  21888  psmetutop  22312  xrge0tsms  22577  fmcfil  23010  ellimc2  23581  limcflf  23585  xrge0tsmsd  29612  poimirlem23  33103  poimirlem24  33104  poimirlem30  33110  cntotbnd  33266
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