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Theorem rexrnmpo 7280
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 321 . . . 4 (𝑧 = 𝐶 → (¬ 𝜑 ↔ ¬ 𝜓))
41, 3ralrnmpo 7279 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∀𝑧 ∈ ran 𝐹 ¬ 𝜑 ↔ ∀𝑥𝐴𝑦𝐵 ¬ 𝜓))
54notbid 321 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (¬ ∀𝑧 ∈ ran 𝐹 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜓))
6 dfrex2 3234 . 2 (∃𝑧 ∈ ran 𝐹𝜑 ↔ ¬ ∀𝑧 ∈ ran 𝐹 ¬ 𝜑)
7 dfrex2 3234 . . . 4 (∃𝑦𝐵 𝜓 ↔ ¬ ∀𝑦𝐵 ¬ 𝜓)
87rexbii 3242 . . 3 (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴 ¬ ∀𝑦𝐵 ¬ 𝜓)
9 rexnal 3233 . . 3 (∃𝑥𝐴 ¬ ∀𝑦𝐵 ¬ 𝜓 ↔ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜓)
108, 9bitri 278 . 2 (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜓)
115, 6, 103bitr4g 317 1 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209   = wceq 1538  wcel 2115  wral 3133  wrex 3134  ran crn 5544  cmpo 7148
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 5190  ax-nul 5197  ax-pr 5318
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-v 3482  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-br 5054  df-opab 5116  df-cnv 5551  df-dm 5553  df-rn 5554  df-oprab 7150  df-mpo 7151
This theorem is referenced by:  lsmass  18793  eltx  22171  txrest  22234  txlm  22251  ptrest  34968
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