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Theorem isdilN 36292
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 36291 . . 3 ((𝐾𝐵𝐷𝐴) → (𝐿𝐷) = {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥)})
76eleq2d 2844 . 2 ((𝐾𝐵𝐷𝐴) → (𝐹 ∈ (𝐿𝐷) ↔ 𝐹 ∈ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥)}))
8 fveq1 6445 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
98eqeq1d 2779 . . . . 5 (𝑓 = 𝐹 → ((𝑓𝑥) = 𝑥 ↔ (𝐹𝑥) = 𝑥))
109imbi2d 332 . . . 4 (𝑓 = 𝐹 → ((𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥) ↔ (𝑥 ⊆ (𝑊𝐷) → (𝐹𝑥) = 𝑥)))
1110ralbidv 3167 . . 3 (𝑓 = 𝐹 → (∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥) ↔ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝐹𝑥) = 𝑥)))
1211elrab 3571 . 2 (𝐹 ∈ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥)} ↔ (𝐹𝑀 ∧ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝐹𝑥) = 𝑥)))
137, 12syl6bb 279 1 ((𝐾𝐵𝐷𝐴) → (𝐹 ∈ (𝐿𝐷) ↔ (𝐹𝑀 ∧ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝐹𝑥) = 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2106  wral 3089  {crab 3093  wss 3791  cfv 6135  Atomscatm 35401  PSubSpcpsubsp 35634  WAtomscwpointsN 36124  PAutcpautN 36125  DilcdilN 36240
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-rep 5006  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-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  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-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-dilN 36244
This theorem is referenced by: (None)
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