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Mirrors > Home > MPE Home > Th. List > pmtrfmvdn0 | Structured version Visualization version GIF version |
Description: A transposition moves at least one point. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
Ref | Expression |
---|---|
pmtrrn.t | ⊢ 𝑇 = (pmTrsp‘𝐷) |
pmtrrn.r | ⊢ 𝑅 = ran 𝑇 |
Ref | Expression |
---|---|
pmtrfmvdn0 | ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2on0 7723 | . 2 ⊢ 2𝑜 ≠ ∅ | |
2 | pmtrrn.t | . . . . . . . 8 ⊢ 𝑇 = (pmTrsp‘𝐷) | |
3 | pmtrrn.r | . . . . . . . 8 ⊢ 𝑅 = ran 𝑇 | |
4 | eqid 2771 | . . . . . . . 8 ⊢ dom (𝐹 ∖ I ) = dom (𝐹 ∖ I ) | |
5 | 2, 3, 4 | pmtrfrn 18085 | . . . . . . 7 ⊢ (𝐹 ∈ 𝑅 → ((𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2𝑜) ∧ 𝐹 = (𝑇‘dom (𝐹 ∖ I )))) |
6 | 5 | simpld 482 | . . . . . 6 ⊢ (𝐹 ∈ 𝑅 → (𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2𝑜)) |
7 | 6 | simp3d 1138 | . . . . 5 ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≈ 2𝑜) |
8 | enen1 8256 | . . . . 5 ⊢ (dom (𝐹 ∖ I ) ≈ 2𝑜 → (dom (𝐹 ∖ I ) ≈ ∅ ↔ 2𝑜 ≈ ∅)) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝐹 ∈ 𝑅 → (dom (𝐹 ∖ I ) ≈ ∅ ↔ 2𝑜 ≈ ∅)) |
10 | en0 8172 | . . . 4 ⊢ (dom (𝐹 ∖ I ) ≈ ∅ ↔ dom (𝐹 ∖ I ) = ∅) | |
11 | en0 8172 | . . . 4 ⊢ (2𝑜 ≈ ∅ ↔ 2𝑜 = ∅) | |
12 | 9, 10, 11 | 3bitr3g 302 | . . 3 ⊢ (𝐹 ∈ 𝑅 → (dom (𝐹 ∖ I ) = ∅ ↔ 2𝑜 = ∅)) |
13 | 12 | necon3bid 2987 | . 2 ⊢ (𝐹 ∈ 𝑅 → (dom (𝐹 ∖ I ) ≠ ∅ ↔ 2𝑜 ≠ ∅)) |
14 | 1, 13 | mpbiri 248 | 1 ⊢ (𝐹 ∈ 𝑅 → dom (𝐹 ∖ I ) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 Vcvv 3351 ∖ cdif 3720 ⊆ wss 3723 ∅c0 4063 class class class wbr 4786 I cid 5156 dom cdm 5249 ran crn 5250 ‘cfv 6031 2𝑜c2o 7707 ≈ cen 8106 pmTrspcpmtr 18068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-om 7213 df-1o 7713 df-2o 7714 df-er 7896 df-en 8110 df-fin 8113 df-pmtr 18069 |
This theorem is referenced by: psgnunilem3 18123 |
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