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Mirrors > Home > MPE Home > Th. List > upgrres1lem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for upgrres1 27103. (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 6659 | . . 3 ⊢ 𝑉 ∈ V |
3 | 2 | difexi 5196 | . 2 ⊢ (𝑉 ∖ {𝑁}) ∈ V |
4 | upgrres1.f | . . . 4 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} | |
5 | upgrres1.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
6 | 5 | fvexi 6659 | . . . 4 ⊢ 𝐸 ∈ V |
7 | 4, 6 | rabex2 5201 | . . 3 ⊢ 𝐹 ∈ V |
8 | resiexg 7601 | . . 3 ⊢ (𝐹 ∈ V → ( I ↾ 𝐹) ∈ V) | |
9 | 7, 8 | ax-mp 5 | . 2 ⊢ ( I ↾ 𝐹) ∈ V |
10 | 3, 9 | pm3.2i 474 | 1 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∉ wnel 3091 {crab 3110 Vcvv 3441 ∖ cdif 3878 {csn 4525 I cid 5424 ↾ cres 5521 ‘cfv 6324 Vtxcvtx 26789 Edgcedg 26840 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-res 5531 df-iota 6283 df-fv 6332 |
This theorem is referenced by: upgrres1lem2 27101 upgrres1lem3 27102 |
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