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Mirrors > Home > MPE Home > Th. List > cusgrexilem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for cusgrexi 27152. (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 5270 | . . 3 ⊢ (𝑉 ∈ 𝑊 → 𝒫 𝑉 ∈ V) | |
3 | 1, 2 | rabexd 5227 | . 2 ⊢ (𝑉 ∈ 𝑊 → 𝑃 ∈ V) |
4 | resiexg 7608 | . 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 1528 ∈ wcel 2105 {crab 3139 Vcvv 3492 𝒫 cpw 4535 I cid 5452 ↾ cres 5550 ‘cfv 6348 2c2 11680 ♯chash 13678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-res 5560 |
This theorem is referenced by: usgrexi 27150 cusgrexi 27152 cusgrexg 27153 structtousgr 27154 structtocusgr 27155 |
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