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Theorem rexrnmpt 7117
Description: A restricted quantifier over an image set. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker rexrnmptw 7115 when possible. (Contributed by Mario Carneiro, 20-Aug-2015.) (New usage is discouraged.)
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 318 . . . 4 (𝑦 = 𝐵 → (¬ 𝜓 ↔ ¬ 𝜒))
41, 3ralrnmpt 7116 . . 3 (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹 ¬ 𝜓 ↔ ∀𝑥𝐴 ¬ 𝜒))
54notbid 318 . 2 (∀𝑥𝐴 𝐵𝑉 → (¬ ∀𝑦 ∈ ran 𝐹 ¬ 𝜓 ↔ ¬ ∀𝑥𝐴 ¬ 𝜒))
6 dfrex2 3073 . 2 (∃𝑦 ∈ ran 𝐹𝜓 ↔ ¬ ∀𝑦 ∈ ran 𝐹 ¬ 𝜓)
7 dfrex2 3073 . 2 (∃𝑥𝐴 𝜒 ↔ ¬ ∀𝑥𝐴 ¬ 𝜒)
85, 6, 73bitr4g 314 1 (∀𝑥𝐴 𝐵𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cmpt 5225  ran crn 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2377  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by: (None)
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