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Theorem pmtridf1o 33097
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 4537 . . . . 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 2787 . . 3 ((𝜑𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
5 f1oi 6887 . . . 4 ( I ↾ 𝐴):𝐴1-1-onto𝐴
65a1i 11 . . 3 ((𝜑𝑋 = 𝑌) → ( I ↾ 𝐴):𝐴1-1-onto𝐴)
7 f1oeq1 6837 . . . 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 584 . 2 ((𝜑𝑋 = 𝑌) → 𝑇:𝐴1-1-onto𝐴)
10 simpr 484 . . . . . . 7 ((𝜑𝑋𝑌) → 𝑋𝑌)
1110neneqd 2943 . . . . . 6 ((𝜑𝑋𝑌) → ¬ 𝑋 = 𝑌)
12 iffalse 4540 . . . . . 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 2787 . . . 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 4827 . . . . 5 ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ 𝐴)
22 enpr2 10040 . . . . . 6 ((𝑋𝐴𝑌𝐴𝑋𝑌) → {𝑋, 𝑌} ≈ 2o)
2318, 20, 10, 22syl3anc 1370 . . . . 5 ((𝜑𝑋𝑌) → {𝑋, 𝑌} ≈ 2o)
24 eqid 2735 . . . . . 6 (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
25 eqid 2735 . . . . . 6 ran (pmTrsp‘𝐴) = ran (pmTrsp‘𝐴)
2624, 25pmtrrn 19490 . . . . 5 ((𝐴𝑉 ∧ {𝑋, 𝑌} ⊆ 𝐴 ∧ {𝑋, 𝑌} ≈ 2o) → ((pmTrsp‘𝐴)‘{𝑋, 𝑌}) ∈ ran (pmTrsp‘𝐴))
2716, 21, 23, 26syl3anc 1370 . . . 4 ((𝜑𝑋𝑌) → ((pmTrsp‘𝐴)‘{𝑋, 𝑌}) ∈ ran (pmTrsp‘𝐴))
2814, 27eqeltrd 2839 . . 3 ((𝜑𝑋𝑌) → 𝑇 ∈ ran (pmTrsp‘𝐴))
2924, 25pmtrff1o 19496 . . 3 (𝑇 ∈ ran (pmTrsp‘𝐴) → 𝑇:𝐴1-1-onto𝐴)
3028, 29syl 17 . 2 ((𝜑𝑋𝑌) → 𝑇:𝐴1-1-onto𝐴)
319, 30pm2.61dane 3027 1 (𝜑𝑇:𝐴1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  wne 2938  wss 3963  ifcif 4531  {cpr 4633   class class class wbr 5148   I cid 5582  ran crn 5690  cres 5691  1-1-ontowf1o 6562  cfv 6563  2oc2o 8499  cen 8981  pmTrspcpmtr 19474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-2o 8506  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pmtr 19475
This theorem is referenced by:  reprpmtf1o  34620  hgt750lema  34651
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