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Theorem pmtridf1o 31992
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 4493 . . . . 5 (𝑋 = 𝑌 → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
32adantl 483 . . . 4 ((𝜑𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
41, 3eqtrid 2785 . . 3 ((𝜑𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
5 f1oi 6823 . . . 4 ( I ↾ 𝐴):𝐴1-1-onto𝐴
65a1i 11 . . 3 ((𝜑𝑋 = 𝑌) → ( I ↾ 𝐴):𝐴1-1-onto𝐴)
7 f1oeq1 6773 . . . 4 (𝑇 = ( I ↾ 𝐴) → (𝑇:𝐴1-1-onto𝐴 ↔ ( I ↾ 𝐴):𝐴1-1-onto𝐴))
87biimpar 479 . . 3 ((𝑇 = ( I ↾ 𝐴) ∧ ( I ↾ 𝐴):𝐴1-1-onto𝐴) → 𝑇:𝐴1-1-onto𝐴)
94, 6, 8syl2anc 585 . 2 ((𝜑𝑋 = 𝑌) → 𝑇:𝐴1-1-onto𝐴)
10 simpr 486 . . . . . . 7 ((𝜑𝑋𝑌) → 𝑋𝑌)
1110neneqd 2945 . . . . . 6 ((𝜑𝑋𝑌) → ¬ 𝑋 = 𝑌)
12 iffalse 4496 . . . . . 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 2785 . . . 4 ((𝜑𝑋𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
15 pmtridf1o.a . . . . . 6 (𝜑𝐴𝑉)
1615adantr 482 . . . . 5 ((𝜑𝑋𝑌) → 𝐴𝑉)
17 pmtridf1o.x . . . . . . 7 (𝜑𝑋𝐴)
1817adantr 482 . . . . . 6 ((𝜑𝑋𝑌) → 𝑋𝐴)
19 pmtridf1o.y . . . . . . 7 (𝜑𝑌𝐴)
2019adantr 482 . . . . . 6 ((𝜑𝑋𝑌) → 𝑌𝐴)
2118, 20prssd 4783 . . . . 5 ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ 𝐴)
22 enpr2 9943 . . . . . 6 ((𝑋𝐴𝑌𝐴𝑋𝑌) → {𝑋, 𝑌} ≈ 2o)
2318, 20, 10, 22syl3anc 1372 . . . . 5 ((𝜑𝑋𝑌) → {𝑋, 𝑌} ≈ 2o)
24 eqid 2733 . . . . . 6 (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
25 eqid 2733 . . . . . 6 ran (pmTrsp‘𝐴) = ran (pmTrsp‘𝐴)
2624, 25pmtrrn 19244 . . . . 5 ((𝐴𝑉 ∧ {𝑋, 𝑌} ⊆ 𝐴 ∧ {𝑋, 𝑌} ≈ 2o) → ((pmTrsp‘𝐴)‘{𝑋, 𝑌}) ∈ ran (pmTrsp‘𝐴))
2716, 21, 23, 26syl3anc 1372 . . . 4 ((𝜑𝑋𝑌) → ((pmTrsp‘𝐴)‘{𝑋, 𝑌}) ∈ ran (pmTrsp‘𝐴))
2814, 27eqeltrd 2834 . . 3 ((𝜑𝑋𝑌) → 𝑇 ∈ ran (pmTrsp‘𝐴))
2924, 25pmtrff1o 19250 . . 3 (𝑇 ∈ ran (pmTrsp‘𝐴) → 𝑇:𝐴1-1-onto𝐴)
3028, 29syl 17 . 2 ((𝜑𝑋𝑌) → 𝑇:𝐴1-1-onto𝐴)
319, 30pm2.61dane 3029 1 (𝜑𝑇:𝐴1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2940  wss 3911  ifcif 4487  {cpr 4589   class class class wbr 5106   I cid 5531  ran crn 5635  cres 5636  1-1-ontowf1o 6496  cfv 6497  2oc2o 8407  cen 8883  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-3or 1089  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-pss 3930  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-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  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-ord 6321  df-on 6322  df-lim 6323  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-om 7804  df-1o 8413  df-2o 8414  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-pmtr 19229
This theorem is referenced by:  reprpmtf1o  33296  hgt750lema  33327
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