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Theorem pmtridf1o 30663
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 4469 . . . . 5 (𝑋 = 𝑌 → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
32adantl 482 . . . 4 ((𝜑𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
41, 3syl5eq 2865 . . 3 ((𝜑𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
5 f1oi 6645 . . . 4 ( I ↾ 𝐴):𝐴1-1-onto𝐴
65a1i 11 . . 3 ((𝜑𝑋 = 𝑌) → ( I ↾ 𝐴):𝐴1-1-onto𝐴)
7 f1oeq1 6597 . . . 4 (𝑇 = ( I ↾ 𝐴) → (𝑇:𝐴1-1-onto𝐴 ↔ ( I ↾ 𝐴):𝐴1-1-onto𝐴))
87biimpar 478 . . 3 ((𝑇 = ( I ↾ 𝐴) ∧ ( I ↾ 𝐴):𝐴1-1-onto𝐴) → 𝑇:𝐴1-1-onto𝐴)
94, 6, 8syl2anc 584 . 2 ((𝜑𝑋 = 𝑌) → 𝑇:𝐴1-1-onto𝐴)
10 simpr 485 . . . . . . 7 ((𝜑𝑋𝑌) → 𝑋𝑌)
1110neneqd 3018 . . . . . 6 ((𝜑𝑋𝑌) → ¬ 𝑋 = 𝑌)
12 iffalse 4472 . . . . . 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 2865 . . . 4 ((𝜑𝑋𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
15 pmtridf1o.a . . . . . 6 (𝜑𝐴𝑉)
1615adantr 481 . . . . 5 ((𝜑𝑋𝑌) → 𝐴𝑉)
17 pmtridf1o.x . . . . . . 7 (𝜑𝑋𝐴)
1817adantr 481 . . . . . 6 ((𝜑𝑋𝑌) → 𝑋𝐴)
19 pmtridf1o.y . . . . . . 7 (𝜑𝑌𝐴)
2019adantr 481 . . . . . 6 ((𝜑𝑋𝑌) → 𝑌𝐴)
2118, 20prssd 4747 . . . . 5 ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ 𝐴)
22 pr2nelem 9418 . . . . . 6 ((𝑋𝐴𝑌𝐴𝑋𝑌) → {𝑋, 𝑌} ≈ 2o)
2318, 20, 10, 22syl3anc 1363 . . . . 5 ((𝜑𝑋𝑌) → {𝑋, 𝑌} ≈ 2o)
24 eqid 2818 . . . . . 6 (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
25 eqid 2818 . . . . . 6 ran (pmTrsp‘𝐴) = ran (pmTrsp‘𝐴)
2624, 25pmtrrn 18514 . . . . 5 ((𝐴𝑉 ∧ {𝑋, 𝑌} ⊆ 𝐴 ∧ {𝑋, 𝑌} ≈ 2o) → ((pmTrsp‘𝐴)‘{𝑋, 𝑌}) ∈ ran (pmTrsp‘𝐴))
2716, 21, 23, 26syl3anc 1363 . . . 4 ((𝜑𝑋𝑌) → ((pmTrsp‘𝐴)‘{𝑋, 𝑌}) ∈ ran (pmTrsp‘𝐴))
2814, 27eqeltrd 2910 . . 3 ((𝜑𝑋𝑌) → 𝑇 ∈ ran (pmTrsp‘𝐴))
2924, 25pmtrff1o 18520 . . 3 (𝑇 ∈ ran (pmTrsp‘𝐴) → 𝑇:𝐴1-1-onto𝐴)
3028, 29syl 17 . 2 ((𝜑𝑋𝑌) → 𝑇:𝐴1-1-onto𝐴)
319, 30pm2.61dane 3101 1 (𝜑𝑇:𝐴1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  wne 3013  wss 3933  ifcif 4463  {cpr 4559   class class class wbr 5057   I cid 5452  ran crn 5549  cres 5550  1-1-ontowf1o 6347  cfv 6348  2oc2o 8085  cen 8494  pmTrspcpmtr 18498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7570  df-1o 8091  df-2o 8092  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-pmtr 18499
This theorem is referenced by:  reprpmtf1o  31796  hgt750lema  31827
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