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Theorem pmtridfv1 30744
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 488 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑋 = 𝑌)
32iftrued 4448 . . . . 5 ((𝜑𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
41, 3syl5eq 2868 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
54fveq1d 6645 . . 3 ((𝜑𝑋 = 𝑌) → (𝑇𝑋) = (( I ↾ 𝐴)‘𝑋))
6 pmtridf1o.x . . . . 5 (𝜑𝑋𝐴)
7 fvresi 6908 . . . . 5 (𝑋𝐴 → (( I ↾ 𝐴)‘𝑋) = 𝑋)
86, 7syl 17 . . . 4 (𝜑 → (( I ↾ 𝐴)‘𝑋) = 𝑋)
98adantr 484 . . 3 ((𝜑𝑋 = 𝑌) → (( I ↾ 𝐴)‘𝑋) = 𝑋)
105, 9, 23eqtrd 2860 . 2 ((𝜑𝑋 = 𝑌) → (𝑇𝑋) = 𝑌)
11 simpr 488 . . . . . . 7 ((𝜑𝑋𝑌) → 𝑋𝑌)
1211neneqd 3012 . . . . . 6 ((𝜑𝑋𝑌) → ¬ 𝑋 = 𝑌)
1312iffalsed 4451 . . . . 5 ((𝜑𝑋𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
141, 13syl5eq 2868 . . . 4 ((𝜑𝑋𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
1514fveq1d 6645 . . 3 ((𝜑𝑋𝑌) → (𝑇𝑋) = (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑋))
16 pmtridf1o.a . . . . 5 (𝜑𝐴𝑉)
1716adantr 484 . . . 4 ((𝜑𝑋𝑌) → 𝐴𝑉)
186adantr 484 . . . 4 ((𝜑𝑋𝑌) → 𝑋𝐴)
19 pmtridf1o.y . . . . 5 (𝜑𝑌𝐴)
2019adantr 484 . . . 4 ((𝜑𝑋𝑌) → 𝑌𝐴)
21 eqid 2821 . . . . 5 (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
2221pmtrprfv 18559 . . . 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 2856 . 2 ((𝜑𝑋𝑌) → (𝑇𝑋) = 𝑌)
2510, 24pm2.61dane 3094 1 (𝜑 → (𝑇𝑋) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wne 3007  ifcif 4440  {cpr 4542   I cid 5432  cres 5530  cfv 6328  pmTrspcpmtr 18547
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 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
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 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7556  df-1o 8077  df-2o 8078  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-pmtr 18548
This theorem is referenced by: (None)
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