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Mirrors > Home > MPE Home > Th. List > upgrres1lem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for upgrres1 27969. (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} |
upgrres1.s | ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ |
Ref | Expression |
---|---|
upgrres1lem3 | ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | upgrres1.s | . . 3 ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ | |
2 | 1 | fveq2i 6828 | . 2 ⊢ (iEdg‘𝑆) = (iEdg‘⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩) |
3 | upgrres1.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | upgrres1.e | . . . 4 ⊢ 𝐸 = (Edg‘𝐺) | |
5 | upgrres1.f | . . . 4 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} | |
6 | 3, 4, 5 | upgrres1lem1 27965 | . . 3 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V) |
7 | opiedgfv 27666 | . . 3 ⊢ (((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V) → (iEdg‘⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩) = ( I ↾ 𝐹)) | |
8 | 6, 7 | ax-mp 5 | . 2 ⊢ (iEdg‘⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩) = ( I ↾ 𝐹) |
9 | 2, 8 | eqtri 2764 | 1 ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1540 ∈ wcel 2105 ∉ wnel 3046 {crab 3403 Vcvv 3441 ∖ cdif 3895 {csn 4573 ⟨cop 4579 I cid 5517 ↾ cres 5622 ‘cfv 6479 Vtxcvtx 27655 iEdgciedg 27656 Edgcedg 27706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-iota 6431 df-fun 6481 df-fv 6487 df-2nd 7900 df-iedg 27658 |
This theorem is referenced by: upgrres1 27969 umgrres1 27970 usgrres1 27971 nbupgrres 28020 |
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