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Theorem pmtridfv1 30087
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 479 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑋 = 𝑌)
32iftrued 4202 . . . . 5 ((𝜑𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
41, 3syl5eq 2770 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
54fveq1d 6306 . . 3 ((𝜑𝑋 = 𝑌) → (𝑇𝑋) = (( I ↾ 𝐴)‘𝑋))
6 pmtridf1o.x . . . . 5 (𝜑𝑋𝐴)
7 fvresi 6555 . . . . 5 (𝑋𝐴 → (( I ↾ 𝐴)‘𝑋) = 𝑋)
86, 7syl 17 . . . 4 (𝜑 → (( I ↾ 𝐴)‘𝑋) = 𝑋)
98adantr 472 . . 3 ((𝜑𝑋 = 𝑌) → (( I ↾ 𝐴)‘𝑋) = 𝑋)
105, 9, 23eqtrd 2762 . 2 ((𝜑𝑋 = 𝑌) → (𝑇𝑋) = 𝑌)
11 simpr 479 . . . . . . 7 ((𝜑𝑋𝑌) → 𝑋𝑌)
1211neneqd 2901 . . . . . 6 ((𝜑𝑋𝑌) → ¬ 𝑋 = 𝑌)
1312iffalsed 4205 . . . . 5 ((𝜑𝑋𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
141, 13syl5eq 2770 . . . 4 ((𝜑𝑋𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
1514fveq1d 6306 . . 3 ((𝜑𝑋𝑌) → (𝑇𝑋) = (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑋))
16 pmtridf1o.a . . . . 5 (𝜑𝐴𝑉)
1716adantr 472 . . . 4 ((𝜑𝑋𝑌) → 𝐴𝑉)
186adantr 472 . . . 4 ((𝜑𝑋𝑌) → 𝑋𝐴)
19 pmtridf1o.y . . . . 5 (𝜑𝑌𝐴)
2019adantr 472 . . . 4 ((𝜑𝑋𝑌) → 𝑌𝐴)
21 eqid 2724 . . . . 5 (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
2221pmtrprfv 17994 . . . 4 ((𝐴𝑉 ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑋) = 𝑌)
2317, 18, 20, 11, 22syl13anc 1441 . . 3 ((𝜑𝑋𝑌) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑋) = 𝑌)
2415, 23eqtrd 2758 . 2 ((𝜑𝑋𝑌) → (𝑇𝑋) = 𝑌)
2510, 24pm2.61dane 2983 1 (𝜑 → (𝑇𝑋) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wcel 2103  wne 2896  ifcif 4194  {cpr 4287   I cid 5127  cres 5220  cfv 6001  pmTrspcpmtr 17982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-om 7183  df-1o 7680  df-2o 7681  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-pmtr 17983
This theorem is referenced by: (None)
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