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Theorem cusgrexilem1 29251

Description: Lemma 1 for cusgrexi 29255. (Contributed by Alexander van der Vekens, 12-Jan-2018.)

**Hypothesis**
Ref	Expression
usgrexi.p	⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}

**Assertion**
Ref	Expression
cusgrexilem1	⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃) ∈ V)

Distinct variable group: 𝑥,𝑉 Allowed substitution hints: 𝑃(𝑥) 𝑊(𝑥)

**Proof of Theorem cusgrexilem1**
Step	Hyp	Ref	Expression
1		usgrexi.p	. . 3 ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}
2		pwexg 5378	. . 3 ⊢ (𝑉 ∈ 𝑊 → 𝒫 𝑉 ∈ V)
3	1, 2	rabexd 5335	. 2 ⊢ (𝑉 ∈ 𝑊 → 𝑃 ∈ V)
4		resiexg 7920	. 2 ⊢ (𝑃 ∈ V → ( I ↾ 𝑃) ∈ V)
5	3, 4	syl 17	1 ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃) ∈ V)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {crab 3429 Vcvv 3471 𝒫 cpw 4603 I cid 5575 ↾ cres 5680 ‘cfv 6548 2c2 12297 ♯chash 14321

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-res 5690

This theorem is referenced by: usgrexi 29253 cusgrexi 29255 cusgrexg 29256 structtousgr 29257 structtocusgr 29258

	
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