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Mirrors > Home > MPE Home > Th. List > cusgrfilem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for cusgrfi 27240. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.) |
Ref | Expression |
---|---|
cusgrfi.v | ⊢ 𝑉 = (Vtx‘𝐺) |
cusgrfi.p | ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})} |
cusgrfi.f | ⊢ 𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁}) |
Ref | Expression |
---|---|
cusgrfilem3 | ⊢ (𝑁 ∈ 𝑉 → (𝑉 ∈ Fin ↔ 𝑃 ∈ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | diffi 8750 | . . 3 ⊢ (𝑉 ∈ Fin → (𝑉 ∖ {𝑁}) ∈ Fin) | |
2 | simpr 487 | . . . . . 6 ⊢ ((𝑁 ∈ 𝑉 ∧ ¬ 𝑉 ∈ Fin) → ¬ 𝑉 ∈ Fin) | |
3 | snfi 8594 | . . . . . 6 ⊢ {𝑁} ∈ Fin | |
4 | difinf 8788 | . . . . . 6 ⊢ ((¬ 𝑉 ∈ Fin ∧ {𝑁} ∈ Fin) → ¬ (𝑉 ∖ {𝑁}) ∈ Fin) | |
5 | 2, 3, 4 | sylancl 588 | . . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ ¬ 𝑉 ∈ Fin) → ¬ (𝑉 ∖ {𝑁}) ∈ Fin) |
6 | 5 | ex 415 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (¬ 𝑉 ∈ Fin → ¬ (𝑉 ∖ {𝑁}) ∈ Fin)) |
7 | 6 | con4d 115 | . . 3 ⊢ (𝑁 ∈ 𝑉 → ((𝑉 ∖ {𝑁}) ∈ Fin → 𝑉 ∈ Fin)) |
8 | 1, 7 | impbid2 228 | . 2 ⊢ (𝑁 ∈ 𝑉 → (𝑉 ∈ Fin ↔ (𝑉 ∖ {𝑁}) ∈ Fin)) |
9 | cusgrfi.f | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁}) | |
10 | cusgrfi.v | . . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺) | |
11 | 10 | fvexi 6684 | . . . . . . . 8 ⊢ 𝑉 ∈ V |
12 | 11 | difexi 5232 | . . . . . . 7 ⊢ (𝑉 ∖ {𝑁}) ∈ V |
13 | mptexg 6984 | . . . . . . 7 ⊢ ((𝑉 ∖ {𝑁}) ∈ V → (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁}) ∈ V) | |
14 | 12, 13 | mp1i 13 | . . . . . 6 ⊢ (𝑁 ∈ 𝑉 → (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁}) ∈ V) |
15 | 9, 14 | eqeltrid 2917 | . . . . 5 ⊢ (𝑁 ∈ 𝑉 → 𝐹 ∈ V) |
16 | cusgrfi.p | . . . . . 6 ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})} | |
17 | 10, 16, 9 | cusgrfilem2 27238 | . . . . 5 ⊢ (𝑁 ∈ 𝑉 → 𝐹:(𝑉 ∖ {𝑁})–1-1-onto→𝑃) |
18 | f1oeq1 6604 | . . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓:(𝑉 ∖ {𝑁})–1-1-onto→𝑃 ↔ 𝐹:(𝑉 ∖ {𝑁})–1-1-onto→𝑃)) | |
19 | 15, 17, 18 | spcedv 3599 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → ∃𝑓 𝑓:(𝑉 ∖ {𝑁})–1-1-onto→𝑃) |
20 | bren 8518 | . . . 4 ⊢ ((𝑉 ∖ {𝑁}) ≈ 𝑃 ↔ ∃𝑓 𝑓:(𝑉 ∖ {𝑁})–1-1-onto→𝑃) | |
21 | 19, 20 | sylibr 236 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑉 ∖ {𝑁}) ≈ 𝑃) |
22 | enfi 8734 | . . 3 ⊢ ((𝑉 ∖ {𝑁}) ≈ 𝑃 → ((𝑉 ∖ {𝑁}) ∈ Fin ↔ 𝑃 ∈ Fin)) | |
23 | 21, 22 | syl 17 | . 2 ⊢ (𝑁 ∈ 𝑉 → ((𝑉 ∖ {𝑁}) ∈ Fin ↔ 𝑃 ∈ Fin)) |
24 | 8, 23 | bitrd 281 | 1 ⊢ (𝑁 ∈ 𝑉 → (𝑉 ∈ Fin ↔ 𝑃 ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 {crab 3142 Vcvv 3494 ∖ cdif 3933 𝒫 cpw 4539 {csn 4567 {cpr 4569 class class class wbr 5066 ↦ cmpt 5146 –1-1-onto→wf1o 6354 ‘cfv 6355 ≈ cen 8506 Fincfn 8509 Vtxcvtx 26781 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-fin 8513 |
This theorem is referenced by: cusgrfi 27240 |
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