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Theorem pmtridfv2 30464
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 6708 . . . . 5 (𝑌𝐴 → (( I ↾ 𝐴)‘𝑌) = 𝑌)
31, 2syl 17 . . . 4 (𝜑 → (( I ↾ 𝐴)‘𝑌) = 𝑌)
43adantr 474 . . 3 ((𝜑𝑋 = 𝑌) → (( I ↾ 𝐴)‘𝑌) = 𝑌)
5 pmtridf1o.t . . . . 5 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
6 simpr 479 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑋 = 𝑌)
76iftrued 4315 . . . . 5 ((𝜑𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
85, 7syl5eq 2826 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
98fveq1d 6450 . . 3 ((𝜑𝑋 = 𝑌) → (𝑇𝑌) = (( I ↾ 𝐴)‘𝑌))
104, 9, 63eqtr4d 2824 . 2 ((𝜑𝑋 = 𝑌) → (𝑇𝑌) = 𝑋)
11 simpr 479 . . . . . . 7 ((𝜑𝑋𝑌) → 𝑋𝑌)
1211neneqd 2974 . . . . . 6 ((𝜑𝑋𝑌) → ¬ 𝑋 = 𝑌)
1312iffalsed 4318 . . . . 5 ((𝜑𝑋𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
145, 13syl5eq 2826 . . . 4 ((𝜑𝑋𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
1514fveq1d 6450 . . 3 ((𝜑𝑋𝑌) → (𝑇𝑌) = (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌))
16 pmtridf1o.a . . . . 5 (𝜑𝐴𝑉)
1716adantr 474 . . . 4 ((𝜑𝑋𝑌) → 𝐴𝑉)
18 pmtridf1o.x . . . . 5 (𝜑𝑋𝐴)
1918adantr 474 . . . 4 ((𝜑𝑋𝑌) → 𝑋𝐴)
201adantr 474 . . . 4 ((𝜑𝑋𝑌) → 𝑌𝐴)
21 eqid 2778 . . . . 5 (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
2221pmtrprfv2 30454 . . . 4 ((𝐴𝑉 ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌) = 𝑋)
2317, 19, 20, 11, 22syl13anc 1440 . . 3 ((𝜑𝑋𝑌) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌) = 𝑋)
2415, 23eqtrd 2814 . 2 ((𝜑𝑋𝑌) → (𝑇𝑌) = 𝑋)
2510, 24pm2.61dane 3057 1 (𝜑 → (𝑇𝑌) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  wne 2969  ifcif 4307  {cpr 4400   I cid 5262  cres 5359  cfv 6137  pmTrspcpmtr 18255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-om 7346  df-1o 7845  df-2o 7846  df-er 8028  df-en 8244  df-dom 8245  df-sdom 8246  df-pmtr 18256
This theorem is referenced by:  reprpmtf1o  31314
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