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Theorem pmtridfv1 32725
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 4529 . . . . 5 ((𝜑𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
41, 3eqtrid 2776 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
54fveq1d 6884 . . 3 ((𝜑𝑋 = 𝑌) → (𝑇𝑋) = (( I ↾ 𝐴)‘𝑋))
6 pmtridf1o.x . . . . 5 (𝜑𝑋𝐴)
7 fvresi 7164 . . . . 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 2768 . 2 ((𝜑𝑋 = 𝑌) → (𝑇𝑋) = 𝑌)
11 simpr 484 . . . . . . 7 ((𝜑𝑋𝑌) → 𝑋𝑌)
1211neneqd 2937 . . . . . 6 ((𝜑𝑋𝑌) → ¬ 𝑋 = 𝑌)
1312iffalsed 4532 . . . . 5 ((𝜑𝑋𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
141, 13eqtrid 2776 . . . 4 ((𝜑𝑋𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
1514fveq1d 6884 . . 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 2724 . . . . 5 (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
2221pmtrprfv 19365 . . . 4 ((𝐴𝑉 ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑋) = 𝑌)
2317, 18, 20, 11, 22syl13anc 1369 . . 3 ((𝜑𝑋𝑌) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑋) = 𝑌)
2415, 23eqtrd 2764 . 2 ((𝜑𝑋𝑌) → (𝑇𝑋) = 𝑌)
2510, 24pm2.61dane 3021 1 (𝜑 → (𝑇𝑋) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wne 2932  ifcif 4521  {cpr 4623   I cid 5564  cres 5669  cfv 6534  pmTrspcpmtr 19353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-1o 8462  df-2o 8463  df-en 8937  df-pmtr 19354
This theorem is referenced by: (None)
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