Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pmtridf1o Structured version   Visualization version   GIF version

Theorem pmtridf1o 32680
Description: Transpositions of 𝑋 and 𝑌 (understood to be the identity when 𝑋 = 𝑌), are bijections. (Contributed by Thierry Arnoux, 1-Jan-2022.)
Hypotheses
Ref Expression
pmtridf1o.a (𝜑𝐴𝑉)
pmtridf1o.x (𝜑𝑋𝐴)
pmtridf1o.y (𝜑𝑌𝐴)
pmtridf1o.t 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
Assertion
Ref Expression
pmtridf1o (𝜑𝑇:𝐴1-1-onto𝐴)

Proof of Theorem pmtridf1o
StepHypRef Expression
1 pmtridf1o.t . . . 4 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
2 iftrue 4526 . . . . 5 (𝑋 = 𝑌 → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
32adantl 481 . . . 4 ((𝜑𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
41, 3eqtrid 2776 . . 3 ((𝜑𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
5 f1oi 6861 . . . 4 ( I ↾ 𝐴):𝐴1-1-onto𝐴
65a1i 11 . . 3 ((𝜑𝑋 = 𝑌) → ( I ↾ 𝐴):𝐴1-1-onto𝐴)
7 f1oeq1 6811 . . . 4 (𝑇 = ( I ↾ 𝐴) → (𝑇:𝐴1-1-onto𝐴 ↔ ( I ↾ 𝐴):𝐴1-1-onto𝐴))
87biimpar 477 . . 3 ((𝑇 = ( I ↾ 𝐴) ∧ ( I ↾ 𝐴):𝐴1-1-onto𝐴) → 𝑇:𝐴1-1-onto𝐴)
94, 6, 8syl2anc 583 . 2 ((𝜑𝑋 = 𝑌) → 𝑇:𝐴1-1-onto𝐴)
10 simpr 484 . . . . . . 7 ((𝜑𝑋𝑌) → 𝑋𝑌)
1110neneqd 2937 . . . . . 6 ((𝜑𝑋𝑌) → ¬ 𝑋 = 𝑌)
12 iffalse 4529 . . . . . 6 𝑋 = 𝑌 → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
1311, 12syl 17 . . . . 5 ((𝜑𝑋𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
141, 13eqtrid 2776 . . . 4 ((𝜑𝑋𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
15 pmtridf1o.a . . . . . 6 (𝜑𝐴𝑉)
1615adantr 480 . . . . 5 ((𝜑𝑋𝑌) → 𝐴𝑉)
17 pmtridf1o.x . . . . . . 7 (𝜑𝑋𝐴)
1817adantr 480 . . . . . 6 ((𝜑𝑋𝑌) → 𝑋𝐴)
19 pmtridf1o.y . . . . . . 7 (𝜑𝑌𝐴)
2019adantr 480 . . . . . 6 ((𝜑𝑋𝑌) → 𝑌𝐴)
2118, 20prssd 4817 . . . . 5 ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ 𝐴)
22 enpr2 9992 . . . . . 6 ((𝑋𝐴𝑌𝐴𝑋𝑌) → {𝑋, 𝑌} ≈ 2o)
2318, 20, 10, 22syl3anc 1368 . . . . 5 ((𝜑𝑋𝑌) → {𝑋, 𝑌} ≈ 2o)
24 eqid 2724 . . . . . 6 (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
25 eqid 2724 . . . . . 6 ran (pmTrsp‘𝐴) = ran (pmTrsp‘𝐴)
2624, 25pmtrrn 19362 . . . . 5 ((𝐴𝑉 ∧ {𝑋, 𝑌} ⊆ 𝐴 ∧ {𝑋, 𝑌} ≈ 2o) → ((pmTrsp‘𝐴)‘{𝑋, 𝑌}) ∈ ran (pmTrsp‘𝐴))
2716, 21, 23, 26syl3anc 1368 . . . 4 ((𝜑𝑋𝑌) → ((pmTrsp‘𝐴)‘{𝑋, 𝑌}) ∈ ran (pmTrsp‘𝐴))
2814, 27eqeltrd 2825 . . 3 ((𝜑𝑋𝑌) → 𝑇 ∈ ran (pmTrsp‘𝐴))
2924, 25pmtrff1o 19368 . . 3 (𝑇 ∈ ran (pmTrsp‘𝐴) → 𝑇:𝐴1-1-onto𝐴)
3028, 29syl 17 . 2 ((𝜑𝑋𝑌) → 𝑇:𝐴1-1-onto𝐴)
319, 30pm2.61dane 3021 1 (𝜑𝑇:𝐴1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1533  wcel 2098  wne 2932  wss 3940  ifcif 4520  {cpr 4622   class class class wbr 5138   I cid 5563  ran crn 5667  cres 5668  1-1-ontowf1o 6532  cfv 6533  2oc2o 8455  cen 8931  pmTrspcpmtr 19346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-om 7849  df-1o 8461  df-2o 8462  df-er 8698  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-pmtr 19347
This theorem is referenced by:  reprpmtf1o  34093  hgt750lema  34124
  Copyright terms: Public domain W3C validator