Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cusgrexilem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for cusgrexi 27219. (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 5272 | . . 3 ⊢ (𝑉 ∈ 𝑊 → 𝒫 𝑉 ∈ V) | |
3 | 1, 2 | rabexd 5229 | . 2 ⊢ (𝑉 ∈ 𝑊 → 𝑃 ∈ V) |
4 | resiexg 7613 | . 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 1533 ∈ wcel 2110 {crab 3142 Vcvv 3495 𝒫 cpw 4539 I cid 5454 ↾ cres 5552 ‘cfv 6350 2c2 11686 ♯chash 13684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-res 5562 |
This theorem is referenced by: usgrexi 27217 cusgrexi 27219 cusgrexg 27220 structtousgr 27221 structtocusgr 27222 |
Copyright terms: Public domain | W3C validator |