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Mirrors > Home > MPE Home > Th. List > cusgrexilem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for cusgrexi 26574. (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 4981 | . . 3 ⊢ (𝑉 ∈ 𝑊 → 𝒫 𝑉 ∈ V) | |
3 | 1, 2 | rabexd 4948 | . 2 ⊢ (𝑉 ∈ 𝑊 → 𝑃 ∈ V) |
4 | resiexg 7253 | . 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 1631 ∈ wcel 2145 {crab 3065 Vcvv 3351 𝒫 cpw 4298 I cid 5157 ↾ cres 5252 ‘cfv 6030 2c2 11276 ♯chash 13321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-id 5158 df-xp 5256 df-rel 5257 df-res 5262 |
This theorem is referenced by: usgrexi 26572 cusgrexi 26574 cusgrexg 26575 structtousgr 26576 structtocusgr 26577 |
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