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Mirrors > Home > MPE Home > Th. List > upgrres1lem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for upgrres1 26375. (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 | fvex 6350 | . . . 4 ⊢ (Vtx‘𝐺) ∈ V | |
3 | 1, 2 | eqeltri 2823 | . . 3 ⊢ 𝑉 ∈ V |
4 | 3 | difexi 4949 | . 2 ⊢ (𝑉 ∖ {𝑁}) ∈ V |
5 | upgrres1.f | . . . 4 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} | |
6 | upgrres1.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
7 | fvex 6350 | . . . . 5 ⊢ (Edg‘𝐺) ∈ V | |
8 | 6, 7 | eqeltri 2823 | . . . 4 ⊢ 𝐸 ∈ V |
9 | 5, 8 | rabex2 4954 | . . 3 ⊢ 𝐹 ∈ V |
10 | resiexg 7255 | . . 3 ⊢ (𝐹 ∈ V → ( I ↾ 𝐹) ∈ V) | |
11 | 9, 10 | ax-mp 5 | . 2 ⊢ ( I ↾ 𝐹) ∈ V |
12 | 4, 11 | pm3.2i 470 | 1 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1620 ∈ wcel 2127 ∉ wnel 3023 {crab 3042 Vcvv 3328 ∖ cdif 3700 {csn 4309 I cid 5161 ↾ cres 5256 ‘cfv 6037 Vtxcvtx 26044 Edgcedg 26109 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ral 3043 df-rex 3044 df-rab 3047 df-v 3330 df-sbc 3565 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-op 4316 df-uni 4577 df-br 4793 df-opab 4853 df-id 5162 df-xp 5260 df-rel 5261 df-res 5266 df-iota 6000 df-fv 6045 |
This theorem is referenced by: upgrres1lem2 26373 upgrres1lem3 26374 |
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