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Theorem pmtridf1o 30754
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 4454 . . . . 5 (𝑋 = 𝑌 → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
32adantl 485 . . . 4 ((𝜑𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
41, 3syl5eq 2871 . . 3 ((𝜑𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
5 f1oi 6633 . . . 4 ( I ↾ 𝐴):𝐴1-1-onto𝐴
65a1i 11 . . 3 ((𝜑𝑋 = 𝑌) → ( I ↾ 𝐴):𝐴1-1-onto𝐴)
7 f1oeq1 6585 . . . 4 (𝑇 = ( I ↾ 𝐴) → (𝑇:𝐴1-1-onto𝐴 ↔ ( I ↾ 𝐴):𝐴1-1-onto𝐴))
87biimpar 481 . . 3 ((𝑇 = ( I ↾ 𝐴) ∧ ( I ↾ 𝐴):𝐴1-1-onto𝐴) → 𝑇:𝐴1-1-onto𝐴)
94, 6, 8syl2anc 587 . 2 ((𝜑𝑋 = 𝑌) → 𝑇:𝐴1-1-onto𝐴)
10 simpr 488 . . . . . . 7 ((𝜑𝑋𝑌) → 𝑋𝑌)
1110neneqd 3018 . . . . . 6 ((𝜑𝑋𝑌) → ¬ 𝑋 = 𝑌)
12 iffalse 4457 . . . . . 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, 13syl5eq 2871 . . . 4 ((𝜑𝑋𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
15 pmtridf1o.a . . . . . 6 (𝜑𝐴𝑉)
1615adantr 484 . . . . 5 ((𝜑𝑋𝑌) → 𝐴𝑉)
17 pmtridf1o.x . . . . . . 7 (𝜑𝑋𝐴)
1817adantr 484 . . . . . 6 ((𝜑𝑋𝑌) → 𝑋𝐴)
19 pmtridf1o.y . . . . . . 7 (𝜑𝑌𝐴)
2019adantr 484 . . . . . 6 ((𝜑𝑋𝑌) → 𝑌𝐴)
2118, 20prssd 4736 . . . . 5 ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ 𝐴)
22 pr2nelem 9415 . . . . . 6 ((𝑋𝐴𝑌𝐴𝑋𝑌) → {𝑋, 𝑌} ≈ 2o)
2318, 20, 10, 22syl3anc 1368 . . . . 5 ((𝜑𝑋𝑌) → {𝑋, 𝑌} ≈ 2o)
24 eqid 2824 . . . . . 6 (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
25 eqid 2824 . . . . . 6 ran (pmTrsp‘𝐴) = ran (pmTrsp‘𝐴)
2624, 25pmtrrn 18574 . . . . 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 2916 . . 3 ((𝜑𝑋𝑌) → 𝑇 ∈ ran (pmTrsp‘𝐴))
2924, 25pmtrff1o 18580 . . 3 (𝑇 ∈ ran (pmTrsp‘𝐴) → 𝑇:𝐴1-1-onto𝐴)
3028, 29syl 17 . 2 ((𝜑𝑋𝑌) → 𝑇:𝐴1-1-onto𝐴)
319, 30pm2.61dane 3100 1 (𝜑𝑇:𝐴1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2115  wne 3013  wss 3918  ifcif 4448  {cpr 4550   class class class wbr 5047   I cid 5440  ran crn 5537  cres 5538  1-1-ontowf1o 6335  cfv 6336  2oc2o 8079  cen 8489  pmTrspcpmtr 18558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-om 7564  df-1o 8085  df-2o 8086  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-pmtr 18559
This theorem is referenced by:  reprpmtf1o  31915  hgt750lema  31946
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