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Theorem rexrnmpo 7500
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 7499 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∀𝑧 ∈ ran 𝐹 ¬ 𝜑 ↔ ∀𝑥𝐴𝑦𝐵 ¬ 𝜓))
54notbid 318 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (¬ ∀𝑧 ∈ ran 𝐹 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜓))
6 dfrex2 3077 . 2 (∃𝑧 ∈ ran 𝐹𝜑 ↔ ¬ ∀𝑧 ∈ ran 𝐹 ¬ 𝜑)
7 dfrex2 3077 . . . 4 (∃𝑦𝐵 𝜓 ↔ ¬ ∀𝑦𝐵 ¬ 𝜓)
87rexbii 3098 . . 3 (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴 ¬ ∀𝑦𝐵 ¬ 𝜓)
9 rexnal 3104 . . 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 205   = wceq 1542  wcel 2107  wral 3065  wrex 3074  ran crn 5639  cmpo 7364
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-cnv 5646  df-dm 5648  df-rn 5649  df-oprab 7366  df-mpo 7367
This theorem is referenced by:  lsmass  19458  eltx  22935  txrest  22998  txlm  23015  lsmssass  32223  ptrest  36106
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