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Theorem rexrnmpt2 7053
 Description: A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)
Hypotheses
Ref Expression
rngop.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
ralrnmpt2.2 (𝑧 = 𝐶 → (𝜑𝜓))
Assertion
Ref Expression
rexrnmpt2 (∀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 rexrnmpt2
StepHypRef Expression
1 rngop.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
2 ralrnmpt2.2 . . . . 5 (𝑧 = 𝐶 → (𝜑𝜓))
32notbid 310 . . . 4 (𝑧 = 𝐶 → (¬ 𝜑 ↔ ¬ 𝜓))
41, 3ralrnmpt2 7052 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∀𝑧 ∈ ran 𝐹 ¬ 𝜑 ↔ ∀𝑥𝐴𝑦𝐵 ¬ 𝜓))
54notbid 310 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (¬ ∀𝑧 ∈ ran 𝐹 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜓))
6 dfrex2 3176 . 2 (∃𝑧 ∈ ran 𝐹𝜑 ↔ ¬ ∀𝑧 ∈ ran 𝐹 ¬ 𝜑)
7 dfrex2 3176 . . . 4 (∃𝑦𝐵 𝜓 ↔ ¬ ∀𝑦𝐵 ¬ 𝜓)
87rexbii 3223 . . 3 (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴 ¬ ∀𝑦𝐵 ¬ 𝜓)
9 rexnal 3175 . . 3 (∃𝑥𝐴 ¬ ∀𝑦𝐵 ¬ 𝜓 ↔ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜓)
108, 9bitri 267 . 2 (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜓)
115, 6, 103bitr4g 306 1 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   = wceq 1601   ∈ wcel 2106  ∀wral 3089  ∃wrex 3090  ran crn 5356   ↦ cmpt2 6924 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4887  df-opab 4949  df-cnv 5363  df-dm 5365  df-rn 5366  df-oprab 6926  df-mpt2 6927 This theorem is referenced by:  lsmass  18467  eltx  21780  txrest  21843  txlm  21860  ptrest  34018
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