![]() |
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 29373. (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 5372 | . . 3 ⊢ (𝑉 ∈ 𝑊 → 𝒫 𝑉 ∈ V) | |
3 | 1, 2 | rabexd 5330 | . 2 ⊢ (𝑉 ∈ 𝑊 → 𝑃 ∈ V) |
4 | resiexg 7914 | . 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 3419 Vcvv 3462 𝒫 cpw 4597 I cid 5569 ↾ cres 5674 ‘cfv 6543 2c2 12310 ♯chash 14339 |
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 2697 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5144 df-opab 5206 df-id 5570 df-xp 5678 df-rel 5679 df-res 5684 |
This theorem is referenced by: usgrexi 29371 cusgrexi 29373 cusgrexg 29374 structtousgr 29375 structtocusgr 29376 |
Copyright terms: Public domain | W3C validator |