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Theorem rexrnmpt 7098
Description: A restricted quantifier over an image set. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker rexrnmptw 7096 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 317 . . . 4 (𝑦 = 𝐵 → (¬ 𝜓 ↔ ¬ 𝜒))
41, 3ralrnmpt 7097 . . 3 (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹 ¬ 𝜓 ↔ ∀𝑥𝐴 ¬ 𝜒))
54notbid 317 . 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 313 1 (∀𝑥𝐴 𝐵𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1541  wcel 2106  wral 3061  wrex 3070  cmpt 5231  ran crn 5677
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2371  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551
This theorem is referenced by: (None)
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