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Mirrors > Home > MPE Home > Th. List > upgrres1lem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for upgrres1 28559. (Contributed by AV, 7-Nov-2020.) |
Ref | Expression |
---|---|
upgrres1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
upgrres1.e | ⊢ 𝐸 = (Edg‘𝐺) |
upgrres1.f | ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} |
Ref | Expression |
---|---|
upgrres1lem1 | ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | upgrres1.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | fvexi 6902 | . . 3 ⊢ 𝑉 ∈ V |
3 | 2 | difexi 5327 | . 2 ⊢ (𝑉 ∖ {𝑁}) ∈ V |
4 | upgrres1.f | . . . 4 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} | |
5 | upgrres1.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
6 | 5 | fvexi 6902 | . . . 4 ⊢ 𝐸 ∈ V |
7 | 4, 6 | rabex2 5333 | . . 3 ⊢ 𝐹 ∈ V |
8 | resiexg 7901 | . . 3 ⊢ (𝐹 ∈ V → ( I ↾ 𝐹) ∈ V) | |
9 | 7, 8 | ax-mp 5 | . 2 ⊢ ( I ↾ 𝐹) ∈ V |
10 | 3, 9 | pm3.2i 471 | 1 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∉ wnel 3046 {crab 3432 Vcvv 3474 ∖ cdif 3944 {csn 4627 I cid 5572 ↾ cres 5677 ‘cfv 6540 Vtxcvtx 28245 Edgcedg 28296 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-res 5687 df-iota 6492 df-fv 6548 |
This theorem is referenced by: upgrres1lem2 28557 upgrres1lem3 28558 |
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