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Theorem ralrnmpt3 41826
 Description: A restricted quantifier over an image set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
ralrnmpt3.1 𝑥𝜑
ralrnmpt3.2 ((𝜑𝑥𝐴) → 𝐵𝑉)
ralrnmpt3.3 (𝑦 = 𝐵 → (𝜓𝜒))
Assertion
Ref Expression
ralrnmpt3 (𝜑 → (∀𝑦 ∈ ran (𝑥𝐴𝐵)𝜓 ↔ ∀𝑥𝐴 𝜒))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝜒,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑥)   𝐵(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem ralrnmpt3
StepHypRef Expression
1 ralrnmpt3.1 . . 3 𝑥𝜑
2 ralrnmpt3.2 . . 3 ((𝜑𝑥𝐴) → 𝐵𝑉)
31, 2ralrimia 41693 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝑉)
4 eqid 2824 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
5 ralrnmpt3.3 . . 3 (𝑦 = 𝐵 → (𝜓𝜒))
64, 5ralrnmptw 6851 . 2 (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran (𝑥𝐴𝐵)𝜓 ↔ ∀𝑥𝐴 𝜒))
73, 6syl 17 1 (𝜑 → (∀𝑦 ∈ ran (𝑥𝐴𝐵)𝜓 ↔ ∀𝑥𝐴 𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2115  ∀wral 3133   ↦ cmpt 5132  ran crn 5543 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-fv 6351 This theorem is referenced by:  liminflelimsuplem  42347
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