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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pmtrprfv2 | Structured version Visualization version GIF version |
Description: In a transposition of two given points, each maps to the other. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
Ref | Expression |
---|---|
pmtrprfv2.t | ⊢ 𝑇 = (pmTrsp‘𝐷) |
Ref | Expression |
---|---|
pmtrprfv2 | ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → ((𝑇‘{𝑋, 𝑌})‘𝑌) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prcom 4694 | . . . 4 ⊢ {𝑌, 𝑋} = {𝑋, 𝑌} | |
2 | 1 | fveq2i 6846 | . . 3 ⊢ (𝑇‘{𝑌, 𝑋}) = (𝑇‘{𝑋, 𝑌}) |
3 | 2 | fveq1i 6844 | . 2 ⊢ ((𝑇‘{𝑌, 𝑋})‘𝑌) = ((𝑇‘{𝑋, 𝑌})‘𝑌) |
4 | ancom 462 | . . . . 5 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷) ↔ (𝑌 ∈ 𝐷 ∧ 𝑋 ∈ 𝐷)) | |
5 | necom 2994 | . . . . 5 ⊢ (𝑋 ≠ 𝑌 ↔ 𝑌 ≠ 𝑋) | |
6 | 4, 5 | anbi12i 628 | . . . 4 ⊢ (((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷) ∧ 𝑋 ≠ 𝑌) ↔ ((𝑌 ∈ 𝐷 ∧ 𝑋 ∈ 𝐷) ∧ 𝑌 ≠ 𝑋)) |
7 | df-3an 1090 | . . . 4 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌) ↔ ((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷) ∧ 𝑋 ≠ 𝑌)) | |
8 | df-3an 1090 | . . . 4 ⊢ ((𝑌 ∈ 𝐷 ∧ 𝑋 ∈ 𝐷 ∧ 𝑌 ≠ 𝑋) ↔ ((𝑌 ∈ 𝐷 ∧ 𝑋 ∈ 𝐷) ∧ 𝑌 ≠ 𝑋)) | |
9 | 6, 7, 8 | 3bitr4i 303 | . . 3 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌) ↔ (𝑌 ∈ 𝐷 ∧ 𝑋 ∈ 𝐷 ∧ 𝑌 ≠ 𝑋)) |
10 | pmtrprfv2.t | . . . 4 ⊢ 𝑇 = (pmTrsp‘𝐷) | |
11 | 10 | pmtrprfv 19240 | . . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ∈ 𝐷 ∧ 𝑌 ≠ 𝑋)) → ((𝑇‘{𝑌, 𝑋})‘𝑌) = 𝑋) |
12 | 9, 11 | sylan2b 595 | . 2 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → ((𝑇‘{𝑌, 𝑋})‘𝑌) = 𝑋) |
13 | 3, 12 | eqtr3id 2787 | 1 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → ((𝑇‘{𝑋, 𝑌})‘𝑌) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 {cpr 4589 ‘cfv 6497 pmTrspcpmtr 19228 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-1o 8413 df-2o 8414 df-en 8887 df-pmtr 19229 |
This theorem is referenced by: pmtrcnel 31989 pmtridfv2 31994 psgnfzto1stlem 31998 |
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