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| Description: Lemma 1 for upgrres1 29330. (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 6920 | . . 3 ⊢ 𝑉 ∈ V | 
| 3 | 2 | difexi 5330 | . 2 ⊢ (𝑉 ∖ {𝑁}) ∈ V | 
| 4 | upgrres1.f | . . . 4 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} | |
| 5 | upgrres1.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
| 6 | 5 | fvexi 6920 | . . . 4 ⊢ 𝐸 ∈ V | 
| 7 | 4, 6 | rabex2 5341 | . . 3 ⊢ 𝐹 ∈ V | 
| 8 | resiexg 7934 | . . 3 ⊢ (𝐹 ∈ V → ( I ↾ 𝐹) ∈ V) | |
| 9 | 7, 8 | ax-mp 5 | . 2 ⊢ ( I ↾ 𝐹) ∈ V | 
| 10 | 3, 9 | pm3.2i 470 | 1 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∉ wnel 3046 {crab 3436 Vcvv 3480 ∖ cdif 3948 {csn 4626 I cid 5577 ↾ cres 5687 ‘cfv 6561 Vtxcvtx 29013 Edgcedg 29064 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-res 5697 df-iota 6514 df-fv 6569 | 
| This theorem is referenced by: upgrres1lem2 29328 upgrres1lem3 29329 | 
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