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Theorem pmtridfv2 30773
Description: Value at Y 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
pmtridfv2 (𝜑 → (𝑇𝑌) = 𝑋)

Proof of Theorem pmtridfv2
StepHypRef Expression
1 pmtridf1o.y . . . . 5 (𝜑𝑌𝐴)
2 fvresi 6926 . . . . 5 (𝑌𝐴 → (( I ↾ 𝐴)‘𝑌) = 𝑌)
31, 2syl 17 . . . 4 (𝜑 → (( I ↾ 𝐴)‘𝑌) = 𝑌)
43adantr 484 . . 3 ((𝜑𝑋 = 𝑌) → (( I ↾ 𝐴)‘𝑌) = 𝑌)
5 pmtridf1o.t . . . . 5 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
6 simpr 488 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑋 = 𝑌)
76iftrued 4458 . . . . 5 ((𝜑𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
85, 7syl5eq 2871 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
98fveq1d 6663 . . 3 ((𝜑𝑋 = 𝑌) → (𝑇𝑌) = (( I ↾ 𝐴)‘𝑌))
104, 9, 63eqtr4d 2869 . 2 ((𝜑𝑋 = 𝑌) → (𝑇𝑌) = 𝑋)
11 simpr 488 . . . . . . 7 ((𝜑𝑋𝑌) → 𝑋𝑌)
1211neneqd 3019 . . . . . 6 ((𝜑𝑋𝑌) → ¬ 𝑋 = 𝑌)
1312iffalsed 4461 . . . . 5 ((𝜑𝑋𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
145, 13syl5eq 2871 . . . 4 ((𝜑𝑋𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
1514fveq1d 6663 . . 3 ((𝜑𝑋𝑌) → (𝑇𝑌) = (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌))
16 pmtridf1o.a . . . . 5 (𝜑𝐴𝑉)
1716adantr 484 . . . 4 ((𝜑𝑋𝑌) → 𝐴𝑉)
18 pmtridf1o.x . . . . 5 (𝜑𝑋𝐴)
1918adantr 484 . . . 4 ((𝜑𝑋𝑌) → 𝑋𝐴)
201adantr 484 . . . 4 ((𝜑𝑋𝑌) → 𝑌𝐴)
21 eqid 2824 . . . . 5 (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
2221pmtrprfv2 30767 . . . 4 ((𝐴𝑉 ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌) = 𝑋)
2317, 19, 20, 11, 22syl13anc 1369 . . 3 ((𝜑𝑋𝑌) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌) = 𝑋)
2415, 23eqtrd 2859 . 2 ((𝜑𝑋𝑌) → (𝑇𝑌) = 𝑋)
2510, 24pm2.61dane 3101 1 (𝜑 → (𝑇𝑌) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wne 3014  ifcif 4450  {cpr 4552   I cid 5446  cres 5544  cfv 6343  pmTrspcpmtr 18569
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 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-om 7575  df-1o 8098  df-2o 8099  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-pmtr 18570
This theorem is referenced by:  reprpmtf1o  31957
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