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Theorem rexrnmpt2 6893
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 307 . . . 4 (𝑧 = 𝐶 → (¬ 𝜑 ↔ ¬ 𝜓))
41, 3ralrnmpt2 6892 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∀𝑧 ∈ ran 𝐹 ¬ 𝜑 ↔ ∀𝑥𝐴𝑦𝐵 ¬ 𝜓))
54notbid 307 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (¬ ∀𝑧 ∈ ran 𝐹 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜓))
6 dfrex2 3098 . 2 (∃𝑧 ∈ ran 𝐹𝜑 ↔ ¬ ∀𝑧 ∈ ran 𝐹 ¬ 𝜑)
7 dfrex2 3098 . . . 4 (∃𝑦𝐵 𝜓 ↔ ¬ ∀𝑦𝐵 ¬ 𝜓)
87rexbii 3143 . . 3 (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴 ¬ ∀𝑦𝐵 ¬ 𝜓)
9 rexnal 3097 . . 3 (∃𝑥𝐴 ¬ ∀𝑦𝐵 ¬ 𝜓 ↔ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜓)
108, 9bitri 264 . 2 (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜓)
115, 6, 103bitr4g 303 1 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1596  wcel 2103  wral 3014  wrex 3015  ran crn 5219  cmpt2 6767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-cnv 5226  df-dm 5228  df-rn 5229  df-oprab 6769  df-mpt2 6770
This theorem is referenced by:  lsmass  18204  eltx  21494  txrest  21557  txlm  21574  ptrest  33640
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