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Mirrors > Home > MPE Home > Th. List > pmtrfv | Structured version Visualization version GIF version |
Description: General value of mapping a point under a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.) |
Ref | Expression |
---|---|
pmtrfval.t | ⊢ 𝑇 = (pmTrsp‘𝐷) |
Ref | Expression |
---|---|
pmtrfv | ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → ((𝑇‘𝑃)‘𝑍) = if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmtrfval.t | . . . . 5 ⊢ 𝑇 = (pmTrsp‘𝐷) | |
2 | 1 | pmtrval 19241 | . . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))) |
3 | 2 | fveq1d 6848 | . . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → ((𝑇‘𝑃)‘𝑍) = ((𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))‘𝑍)) |
4 | 3 | adantr 482 | . 2 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → ((𝑇‘𝑃)‘𝑍) = ((𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))‘𝑍)) |
5 | eqid 2733 | . . 3 ⊢ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)) | |
6 | eleq1 2822 | . . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 ∈ 𝑃 ↔ 𝑍 ∈ 𝑃)) | |
7 | sneq 4600 | . . . . . 6 ⊢ (𝑧 = 𝑍 → {𝑧} = {𝑍}) | |
8 | 7 | difeq2d 4086 | . . . . 5 ⊢ (𝑧 = 𝑍 → (𝑃 ∖ {𝑧}) = (𝑃 ∖ {𝑍})) |
9 | 8 | unieqd 4883 | . . . 4 ⊢ (𝑧 = 𝑍 → ∪ (𝑃 ∖ {𝑧}) = ∪ (𝑃 ∖ {𝑍})) |
10 | id 22 | . . . 4 ⊢ (𝑧 = 𝑍 → 𝑧 = 𝑍) | |
11 | 6, 9, 10 | ifbieq12d 4518 | . . 3 ⊢ (𝑧 = 𝑍 → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) = if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍)) |
12 | simpr 486 | . . 3 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → 𝑍 ∈ 𝐷) | |
13 | simpl3 1194 | . . . . 5 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → 𝑃 ≈ 2o) | |
14 | relen 8894 | . . . . . 6 ⊢ Rel ≈ | |
15 | 14 | brrelex1i 5692 | . . . . 5 ⊢ (𝑃 ≈ 2o → 𝑃 ∈ V) |
16 | difexg 5288 | . . . . 5 ⊢ (𝑃 ∈ V → (𝑃 ∖ {𝑍}) ∈ V) | |
17 | uniexg 7681 | . . . . 5 ⊢ ((𝑃 ∖ {𝑍}) ∈ V → ∪ (𝑃 ∖ {𝑍}) ∈ V) | |
18 | 13, 15, 16, 17 | 4syl 19 | . . . 4 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → ∪ (𝑃 ∖ {𝑍}) ∈ V) |
19 | ifexg 4539 | . . . 4 ⊢ ((∪ (𝑃 ∖ {𝑍}) ∈ V ∧ 𝑍 ∈ 𝐷) → if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍) ∈ V) | |
20 | 18, 19 | sylancom 589 | . . 3 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍) ∈ V) |
21 | 5, 11, 12, 20 | fvmptd3 6975 | . 2 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → ((𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))‘𝑍) = if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍)) |
22 | 4, 21 | eqtrd 2773 | 1 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → ((𝑇‘𝑃)‘𝑍) = if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 Vcvv 3447 ∖ cdif 3911 ⊆ wss 3914 ifcif 4490 {csn 4590 ∪ cuni 4869 class class class wbr 5109 ↦ cmpt 5192 ‘cfv 6500 2oc2o 8410 ≈ cen 8886 pmTrspcpmtr 19231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-en 8890 df-pmtr 19232 |
This theorem is referenced by: pmtrprfv 19243 pmtrprfv3 19244 pmtrmvd 19246 pmtrffv 19249 |
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