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Theorem rexrnmpo 7573
Description: A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)
Hypotheses
Ref Expression
rngop.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
ralrnmpo.2 (𝑧 = 𝐶 → (𝜑𝜓))
Assertion
Ref Expression
rexrnmpo (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓))
Distinct variable groups:   𝑦,𝑧,𝐴   𝑧,𝐵   𝑧,𝐶   𝑧,𝐹   𝜓,𝑧   𝑥,𝑦,𝑧   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑧)   𝜓(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem rexrnmpo
StepHypRef Expression
1 rngop.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
2 ralrnmpo.2 . . . . 5 (𝑧 = 𝐶 → (𝜑𝜓))
32notbid 318 . . . 4 (𝑧 = 𝐶 → (¬ 𝜑 ↔ ¬ 𝜓))
41, 3ralrnmpo 7572 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∀𝑧 ∈ ran 𝐹 ¬ 𝜑 ↔ ∀𝑥𝐴𝑦𝐵 ¬ 𝜓))
54notbid 318 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (¬ ∀𝑧 ∈ ran 𝐹 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜓))
6 dfrex2 3073 . 2 (∃𝑧 ∈ ran 𝐹𝜑 ↔ ¬ ∀𝑧 ∈ ran 𝐹 ¬ 𝜑)
7 dfrex2 3073 . . . 4 (∃𝑦𝐵 𝜓 ↔ ¬ ∀𝑦𝐵 ¬ 𝜓)
87rexbii 3094 . . 3 (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴 ¬ ∀𝑦𝐵 ¬ 𝜓)
9 rexnal 3100 . . 3 (∃𝑥𝐴 ¬ ∀𝑦𝐵 ¬ 𝜓 ↔ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜓)
108, 9bitri 275 . 2 (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜓)
115, 6, 103bitr4g 314 1 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1540  wcel 2108  wral 3061  wrex 3070  ran crn 5686  cmpo 7433
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-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-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-cnv 5693  df-dm 5695  df-rn 5696  df-oprab 7435  df-mpo 7436
This theorem is referenced by:  lsmass  19687  eltx  23576  txrest  23639  txlm  23656  lsmssass  33430  ptrest  37626
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