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Theorem pmtridfv2 29986
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 6480 . . . . 5 (𝑌𝐴 → (( I ↾ 𝐴)‘𝑌) = 𝑌)
31, 2syl 17 . . . 4 (𝜑 → (( I ↾ 𝐴)‘𝑌) = 𝑌)
43adantr 480 . . 3 ((𝜑𝑋 = 𝑌) → (( I ↾ 𝐴)‘𝑌) = 𝑌)
5 pmtridf1o.t . . . . 5 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
6 simpr 476 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑋 = 𝑌)
76iftrued 4127 . . . . 5 ((𝜑𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
85, 7syl5eq 2697 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
98fveq1d 6231 . . 3 ((𝜑𝑋 = 𝑌) → (𝑇𝑌) = (( I ↾ 𝐴)‘𝑌))
104, 9, 63eqtr4d 2695 . 2 ((𝜑𝑋 = 𝑌) → (𝑇𝑌) = 𝑋)
11 simpr 476 . . . . . . 7 ((𝜑𝑋𝑌) → 𝑋𝑌)
1211neneqd 2828 . . . . . 6 ((𝜑𝑋𝑌) → ¬ 𝑋 = 𝑌)
1312iffalsed 4130 . . . . 5 ((𝜑𝑋𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
145, 13syl5eq 2697 . . . 4 ((𝜑𝑋𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
1514fveq1d 6231 . . 3 ((𝜑𝑋𝑌) → (𝑇𝑌) = (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌))
16 pmtridf1o.a . . . . 5 (𝜑𝐴𝑉)
1716adantr 480 . . . 4 ((𝜑𝑋𝑌) → 𝐴𝑉)
18 pmtridf1o.x . . . . 5 (𝜑𝑋𝐴)
1918adantr 480 . . . 4 ((𝜑𝑋𝑌) → 𝑋𝐴)
201adantr 480 . . . 4 ((𝜑𝑋𝑌) → 𝑌𝐴)
21 eqid 2651 . . . . 5 (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
2221pmtrprfv2 29976 . . . 4 ((𝐴𝑉 ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌) = 𝑋)
2317, 19, 20, 11, 22syl13anc 1368 . . 3 ((𝜑𝑋𝑌) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌) = 𝑋)
2415, 23eqtrd 2685 . 2 ((𝜑𝑋𝑌) → (𝑇𝑌) = 𝑋)
2510, 24pm2.61dane 2910 1 (𝜑 → (𝑇𝑌) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wne 2823  ifcif 4119  {cpr 4212   I cid 5052  cres 5145  cfv 5926  pmTrspcpmtr 17907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-2o 7606  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pmtr 17908
This theorem is referenced by:  reprpmtf1o  30832
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