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Theorem isdilN 37395
 Description: The predicate "is a dilation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
dilset.a 𝐴 = (Atoms‘𝐾)
dilset.s 𝑆 = (PSubSp‘𝐾)
dilset.w 𝑊 = (WAtoms‘𝐾)
dilset.m 𝑀 = (PAut‘𝐾)
dilset.l 𝐿 = (Dil‘𝐾)
Assertion
Ref Expression
isdilN ((𝐾𝐵𝐷𝐴) → (𝐹 ∈ (𝐿𝐷) ↔ (𝐹𝑀 ∧ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝐹𝑥) = 𝑥))))
Distinct variable groups:   𝑥,𝐾   𝑥,𝑆   𝑥,𝐷   𝑥,𝐹
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐿(𝑥)   𝑀(𝑥)   𝑊(𝑥)

Proof of Theorem isdilN
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 dilset.a . . . 4 𝐴 = (Atoms‘𝐾)
2 dilset.s . . . 4 𝑆 = (PSubSp‘𝐾)
3 dilset.w . . . 4 𝑊 = (WAtoms‘𝐾)
4 dilset.m . . . 4 𝑀 = (PAut‘𝐾)
5 dilset.l . . . 4 𝐿 = (Dil‘𝐾)
61, 2, 3, 4, 5dilsetN 37394 . . 3 ((𝐾𝐵𝐷𝐴) → (𝐿𝐷) = {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥)})
76eleq2d 2901 . 2 ((𝐾𝐵𝐷𝐴) → (𝐹 ∈ (𝐿𝐷) ↔ 𝐹 ∈ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥)}))
8 fveq1 6660 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
98eqeq1d 2826 . . . . 5 (𝑓 = 𝐹 → ((𝑓𝑥) = 𝑥 ↔ (𝐹𝑥) = 𝑥))
109imbi2d 344 . . . 4 (𝑓 = 𝐹 → ((𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥) ↔ (𝑥 ⊆ (𝑊𝐷) → (𝐹𝑥) = 𝑥)))
1110ralbidv 3192 . . 3 (𝑓 = 𝐹 → (∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥) ↔ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝐹𝑥) = 𝑥)))
1211elrab 3666 . 2 (𝐹 ∈ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥)} ↔ (𝐹𝑀 ∧ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝐹𝑥) = 𝑥)))
137, 12syl6bb 290 1 ((𝐾𝐵𝐷𝐴) → (𝐹 ∈ (𝐿𝐷) ↔ (𝐹𝑀 ∧ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝐹𝑥) = 𝑥))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∀wral 3133  {crab 3137   ⊆ wss 3919  ‘cfv 6343  Atomscatm 36504  PSubSpcpsubsp 36737  WAtomscwpointsN 37227  PAutcpautN 37228  DilcdilN 37343 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-rep 5176  ax-sep 5189  ax-nul 5196  ax-pr 5317 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-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  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-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-dilN 37347 This theorem is referenced by: (None)
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