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Theorem pmtridfv1 31341
Description: Value at X of the transposition of 𝑋 and 𝑌 (understood to be the identity when X = Y ). (Contributed by Thierry Arnoux, 3-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
pmtridfv1 (𝜑 → (𝑇𝑋) = 𝑌)

Proof of Theorem pmtridfv1
StepHypRef Expression
1 pmtridf1o.t . . . . 5 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
2 simpr 484 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑋 = 𝑌)
32iftrued 4472 . . . . 5 ((𝜑𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
41, 3eqtrid 2791 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
54fveq1d 6770 . . 3 ((𝜑𝑋 = 𝑌) → (𝑇𝑋) = (( I ↾ 𝐴)‘𝑋))
6 pmtridf1o.x . . . . 5 (𝜑𝑋𝐴)
7 fvresi 7039 . . . . 5 (𝑋𝐴 → (( I ↾ 𝐴)‘𝑋) = 𝑋)
86, 7syl 17 . . . 4 (𝜑 → (( I ↾ 𝐴)‘𝑋) = 𝑋)
98adantr 480 . . 3 ((𝜑𝑋 = 𝑌) → (( I ↾ 𝐴)‘𝑋) = 𝑋)
105, 9, 23eqtrd 2783 . 2 ((𝜑𝑋 = 𝑌) → (𝑇𝑋) = 𝑌)
11 simpr 484 . . . . . . 7 ((𝜑𝑋𝑌) → 𝑋𝑌)
1211neneqd 2949 . . . . . 6 ((𝜑𝑋𝑌) → ¬ 𝑋 = 𝑌)
1312iffalsed 4475 . . . . 5 ((𝜑𝑋𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
141, 13eqtrid 2791 . . . 4 ((𝜑𝑋𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
1514fveq1d 6770 . . 3 ((𝜑𝑋𝑌) → (𝑇𝑋) = (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑋))
16 pmtridf1o.a . . . . 5 (𝜑𝐴𝑉)
1716adantr 480 . . . 4 ((𝜑𝑋𝑌) → 𝐴𝑉)
186adantr 480 . . . 4 ((𝜑𝑋𝑌) → 𝑋𝐴)
19 pmtridf1o.y . . . . 5 (𝜑𝑌𝐴)
2019adantr 480 . . . 4 ((𝜑𝑋𝑌) → 𝑌𝐴)
21 eqid 2739 . . . . 5 (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
2221pmtrprfv 19042 . . . 4 ((𝐴𝑉 ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑋) = 𝑌)
2317, 18, 20, 11, 22syl13anc 1370 . . 3 ((𝜑𝑋𝑌) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑋) = 𝑌)
2415, 23eqtrd 2779 . 2 ((𝜑𝑋𝑌) → (𝑇𝑋) = 𝑌)
2510, 24pm2.61dane 3033 1 (𝜑 → (𝑇𝑋) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1541  wcel 2109  wne 2944  ifcif 4464  {cpr 4568   I cid 5487  cres 5590  cfv 6430  pmTrspcpmtr 19030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-om 7701  df-1o 8281  df-2o 8282  df-er 8472  df-en 8708  df-dom 8709  df-sdom 8710  df-fin 8711  df-pmtr 19031
This theorem is referenced by: (None)
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