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| Description: Lemma 1 for cusgrexi 29460. (Contributed by Alexander van der Vekens, 12-Jan-2018.) | 
| Ref | Expression | 
|---|---|
| usgrexi.p | ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} | 
| Ref | Expression | 
|---|---|
| cusgrexilem1 | ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃) ∈ V) | 
| 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 5340 | . 2 ⊢ (𝑉 ∈ 𝑊 → 𝑃 ∈ V) | 
| 4 | resiexg 7934 | . 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 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 𝒫 cpw 4600 I cid 5577 ↾ cres 5687 ‘cfv 6561 2c2 12321 ♯chash 14369 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-res 5697 | 
| This theorem is referenced by: usgrexi 29458 cusgrexi 29460 cusgrexg 29461 structtousgr 29462 structtocusgr 29463 | 
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