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Theorem pmtrdifellem3 19268
Description: Lemma 3 for pmtrdifel 19270. (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 19267 . . . . . 6 (𝑄𝑇 → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))
54adantr 482 . . . . 5 ((𝑄𝑇𝑥 ∈ (𝑁 ∖ {𝐾})) → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))
65eleq2d 2820 . . . 4 ((𝑄𝑇𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑥 ∈ dom (𝑆 ∖ I ) ↔ 𝑥 ∈ dom (𝑄 ∖ I )))
74difeq1d 4085 . . . . . 6 (𝑄𝑇 → (dom (𝑆 ∖ I ) ∖ {𝑥}) = (dom (𝑄 ∖ I ) ∖ {𝑥}))
87unieqd 4883 . . . . 5 (𝑄𝑇 (dom (𝑆 ∖ I ) ∖ {𝑥}) = (dom (𝑄 ∖ I ) ∖ {𝑥}))
98adantr 482 . . . 4 ((𝑄𝑇𝑥 ∈ (𝑁 ∖ {𝐾})) → (dom (𝑆 ∖ I ) ∖ {𝑥}) = (dom (𝑄 ∖ I ) ∖ {𝑥}))
106, 9ifbieq1d 4514 . . 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 19266 . . . 4 (𝑄𝑇𝑆𝑅)
12 eldifi 4090 . . . 4 (𝑥 ∈ (𝑁 ∖ {𝐾}) → 𝑥𝑁)
13 eqid 2733 . . . . 5 (pmTrsp‘𝑁) = (pmTrsp‘𝑁)
14 eqid 2733 . . . . 5 dom (𝑆 ∖ I ) = dom (𝑆 ∖ I )
1513, 2, 14pmtrffv 19249 . . . 4 ((𝑆𝑅𝑥𝑁) → (𝑆𝑥) = if(𝑥 ∈ dom (𝑆 ∖ I ), (dom (𝑆 ∖ I ) ∖ {𝑥}), 𝑥))
1611, 12, 15syl2an 597 . . 3 ((𝑄𝑇𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑆𝑥) = if(𝑥 ∈ dom (𝑆 ∖ I ), (dom (𝑆 ∖ I ) ∖ {𝑥}), 𝑥))
17 eqid 2733 . . . 4 (pmTrsp‘(𝑁 ∖ {𝐾})) = (pmTrsp‘(𝑁 ∖ {𝐾}))
18 eqid 2733 . . . 4 dom (𝑄 ∖ I ) = dom (𝑄 ∖ I )
1917, 1, 18pmtrffv 19249 . . 3 ((𝑄𝑇𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑄𝑥) = if(𝑥 ∈ dom (𝑄 ∖ I ), (dom (𝑄 ∖ I ) ∖ {𝑥}), 𝑥))
2010, 16, 193eqtr4rd 2784 . 2 ((𝑄𝑇𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑄𝑥) = (𝑆𝑥))
2120ralrimiva 3140 1 (𝑄𝑇 → ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑄𝑥) = (𝑆𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3061  cdif 3911  ifcif 4490  {csn 4590   cuni 4869   I cid 5534  dom cdm 5637  ran crn 5638  cfv 6500  pmTrspcpmtr 19231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-om 7807  df-1o 8416  df-2o 8417  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-pmtr 19232
This theorem is referenced by:  pmtrdifel  19270  pmtrdifwrdellem3  19273
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