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Theorem pmtrdifellem3 17944
Description: Lemma 3 for pmtrdifel 17946. (Contributed by AV, 15-Jan-2019.)
Hypotheses
Ref Expression
pmtrdifel.t 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
pmtrdifel.r 𝑅 = ran (pmTrsp‘𝑁)
pmtrdifel.0 𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))
Assertion
Ref Expression
pmtrdifellem3 (𝑄𝑇 → ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑄𝑥) = (𝑆𝑥))
Distinct variable groups:   𝑥,𝑄   𝑥,𝑇
Allowed substitution hints:   𝑅(𝑥)   𝑆(𝑥)   𝐾(𝑥)   𝑁(𝑥)

Proof of Theorem pmtrdifellem3
StepHypRef Expression
1 pmtrdifel.t . . . . . . 7 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
2 pmtrdifel.r . . . . . . 7 𝑅 = ran (pmTrsp‘𝑁)
3 pmtrdifel.0 . . . . . . 7 𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))
41, 2, 3pmtrdifellem2 17943 . . . . . 6 (𝑄𝑇 → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))
54adantr 480 . . . . 5 ((𝑄𝑇𝑥 ∈ (𝑁 ∖ {𝐾})) → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))
65eleq2d 2716 . . . 4 ((𝑄𝑇𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑥 ∈ dom (𝑆 ∖ I ) ↔ 𝑥 ∈ dom (𝑄 ∖ I )))
74difeq1d 3760 . . . . . 6 (𝑄𝑇 → (dom (𝑆 ∖ I ) ∖ {𝑥}) = (dom (𝑄 ∖ I ) ∖ {𝑥}))
87unieqd 4478 . . . . 5 (𝑄𝑇 (dom (𝑆 ∖ I ) ∖ {𝑥}) = (dom (𝑄 ∖ I ) ∖ {𝑥}))
98adantr 480 . . . 4 ((𝑄𝑇𝑥 ∈ (𝑁 ∖ {𝐾})) → (dom (𝑆 ∖ I ) ∖ {𝑥}) = (dom (𝑄 ∖ I ) ∖ {𝑥}))
106, 9ifbieq1d 4142 . . 3 ((𝑄𝑇𝑥 ∈ (𝑁 ∖ {𝐾})) → if(𝑥 ∈ dom (𝑆 ∖ I ), (dom (𝑆 ∖ I ) ∖ {𝑥}), 𝑥) = if(𝑥 ∈ dom (𝑄 ∖ I ), (dom (𝑄 ∖ I ) ∖ {𝑥}), 𝑥))
111, 2, 3pmtrdifellem1 17942 . . . 4 (𝑄𝑇𝑆𝑅)
12 eldifi 3765 . . . 4 (𝑥 ∈ (𝑁 ∖ {𝐾}) → 𝑥𝑁)
13 eqid 2651 . . . . 5 (pmTrsp‘𝑁) = (pmTrsp‘𝑁)
14 eqid 2651 . . . . 5 dom (𝑆 ∖ I ) = dom (𝑆 ∖ I )
1513, 2, 14pmtrffv 17925 . . . 4 ((𝑆𝑅𝑥𝑁) → (𝑆𝑥) = if(𝑥 ∈ dom (𝑆 ∖ I ), (dom (𝑆 ∖ I ) ∖ {𝑥}), 𝑥))
1611, 12, 15syl2an 493 . . 3 ((𝑄𝑇𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑆𝑥) = if(𝑥 ∈ dom (𝑆 ∖ I ), (dom (𝑆 ∖ I ) ∖ {𝑥}), 𝑥))
17 eqid 2651 . . . 4 (pmTrsp‘(𝑁 ∖ {𝐾})) = (pmTrsp‘(𝑁 ∖ {𝐾}))
18 eqid 2651 . . . 4 dom (𝑄 ∖ I ) = dom (𝑄 ∖ I )
1917, 1, 18pmtrffv 17925 . . 3 ((𝑄𝑇𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑄𝑥) = if(𝑥 ∈ dom (𝑄 ∖ I ), (dom (𝑄 ∖ I ) ∖ {𝑥}), 𝑥))
2010, 16, 193eqtr4rd 2696 . 2 ((𝑄𝑇𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑄𝑥) = (𝑆𝑥))
2120ralrimiva 2995 1 (𝑄𝑇 → ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑄𝑥) = (𝑆𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  cdif 3604  ifcif 4119  {csn 4210   cuni 4468   I cid 5052  dom cdm 5143  ran crn 5144  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-fin 8001  df-pmtr 17908
This theorem is referenced by:  pmtrdifel  17946  pmtrdifwrdellem3  17949
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