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Theorem cusgrexilem1 26563
Description: Lemma 1 for cusgrexi 26567. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
Hypothesis
Ref Expression
usgrexi.p 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}
Assertion
Ref Expression
cusgrexilem1 (𝑉𝑊 → ( I ↾ 𝑃) ∈ V)
Distinct variable group:   𝑥,𝑉
Allowed substitution hints:   𝑃(𝑥)   𝑊(𝑥)

Proof of Theorem cusgrexilem1
StepHypRef Expression
1 usgrexi.p . . 3 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}
2 pwexg 5048 . . 3 (𝑉𝑊 → 𝒫 𝑉 ∈ V)
31, 2rabexd 5008 . 2 (𝑉𝑊𝑃 ∈ V)
4 resiexg 7332 . 2 (𝑃 ∈ V → ( I ↾ 𝑃) ∈ V)
53, 4syl 17 1 (𝑉𝑊 → ( I ↾ 𝑃) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1637  wcel 2156  {crab 3100  Vcvv 3391  𝒫 cpw 4351   I cid 5218  cres 5313  cfv 6101  2c2 11356  chash 13337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-id 5219  df-xp 5317  df-rel 5318  df-res 5323
This theorem is referenced by:  usgrexi  26565  cusgrexi  26567  cusgrexg  26568  structtousgr  26569  structtocusgr  26570
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