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Theorem pmtridfv2 30192
 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 6582 . . . . 5 (𝑌𝐴 → (( I ↾ 𝐴)‘𝑌) = 𝑌)
31, 2syl 17 . . . 4 (𝜑 → (( I ↾ 𝐴)‘𝑌) = 𝑌)
43adantr 466 . . 3 ((𝜑𝑋 = 𝑌) → (( I ↾ 𝐴)‘𝑌) = 𝑌)
5 pmtridf1o.t . . . . 5 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
6 simpr 471 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑋 = 𝑌)
76iftrued 4231 . . . . 5 ((𝜑𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
85, 7syl5eq 2816 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
98fveq1d 6334 . . 3 ((𝜑𝑋 = 𝑌) → (𝑇𝑌) = (( I ↾ 𝐴)‘𝑌))
104, 9, 63eqtr4d 2814 . 2 ((𝜑𝑋 = 𝑌) → (𝑇𝑌) = 𝑋)
11 simpr 471 . . . . . . 7 ((𝜑𝑋𝑌) → 𝑋𝑌)
1211neneqd 2947 . . . . . 6 ((𝜑𝑋𝑌) → ¬ 𝑋 = 𝑌)
1312iffalsed 4234 . . . . 5 ((𝜑𝑋𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
145, 13syl5eq 2816 . . . 4 ((𝜑𝑋𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
1514fveq1d 6334 . . 3 ((𝜑𝑋𝑌) → (𝑇𝑌) = (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌))
16 pmtridf1o.a . . . . 5 (𝜑𝐴𝑉)
1716adantr 466 . . . 4 ((𝜑𝑋𝑌) → 𝐴𝑉)
18 pmtridf1o.x . . . . 5 (𝜑𝑋𝐴)
1918adantr 466 . . . 4 ((𝜑𝑋𝑌) → 𝑋𝐴)
201adantr 466 . . . 4 ((𝜑𝑋𝑌) → 𝑌𝐴)
21 eqid 2770 . . . . 5 (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
2221pmtrprfv2 30182 . . . 4 ((𝐴𝑉 ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌) = 𝑋)
2317, 19, 20, 11, 22syl13anc 1477 . . 3 ((𝜑𝑋𝑌) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌) = 𝑋)
2415, 23eqtrd 2804 . 2 ((𝜑𝑋𝑌) → (𝑇𝑌) = 𝑋)
2510, 24pm2.61dane 3029 1 (𝜑 → (𝑇𝑌) = 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1630   ∈ wcel 2144   ≠ wne 2942  ifcif 4223  {cpr 4316   I cid 5156   ↾ cres 5251  ‘cfv 6031  pmTrspcpmtr 18067 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7212  df-1o 7712  df-2o 7713  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-pmtr 18068 This theorem is referenced by:  reprpmtf1o  31038
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