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Theorem ralrnmpt3 43836
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 3256 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝑉)
4 eqid 2733 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
5 ralrnmpt3.3 . . 3 (𝑦 = 𝐵 → (𝜓𝜒))
64, 5ralrnmptw 7083 . 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 205  wa 397   = wceq 1542  wnf 1786  wcel 2107  wral 3062  cmpt 5227  ran crn 5673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-fv 6543
This theorem is referenced by:  liminflelimsuplem  44364
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