Theorem cusgraexilem1 25995

Description: Lemma 1 for cusgraexi 25997. (Contributed by Alexander van der Vekens, 12-Jan-2018.)

Hypothesis

Ref	Expression
cusgraexi.p	⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}

Assertion

Ref	Expression
cusgraexilem1	⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃) ∈ V)

Distinct variable group:   𝑥,𝑉

Allowed substitution hints:   𝑃(𝑥)   𝑊(𝑥)

Proof of Theorem cusgraexilem1

Step	Hyp	Ref	Expression
1		cusgraexi.p	. . 3 ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}
2		pwexg 4776	. . 3 ⊢ (𝑉 ∈ 𝑊 → 𝒫 𝑉 ∈ V)
3	1, 2	rabexd 4741	. 2 ⊢ (𝑉 ∈ 𝑊 → 𝑃 ∈ V)
4		resiexg 6994	. 2 ⊢ (𝑃 ∈ V → ( I ↾ 𝑃) ∈ V)
5	3, 4	syl 17	1 ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃) ∈ V)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  𝒫 cpw 4108   I cid 4948   ↾ cres 5040  ‘cfv 5804  2c2 10947  #chash 12979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-res 5050

This theorem is referenced by:  cusgraexilem2  25996  cusgraexi  25997  cusgraexg  25998  cusgrexi  40662  cusgrexg  40663