Theorem pmtrdifellem3 18608

Description: Lemma 3 for pmtrdifel 18610. (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	⊢ (𝑄 ∈ 𝑇 → ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑥) = (𝑆‘𝑥))

Distinct variable groups:   𝑥,𝑄   𝑥,𝑇

Allowed substitution hints:   𝑅(𝑥)   𝑆(𝑥)   𝐾(𝑥)   𝑁(𝑥)

Proof of Theorem pmtrdifellem3

Step	Hyp	Ref	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 ))
4	1, 2, 3	pmtrdifellem2 18607	. . . . . 6 ⊢ (𝑄 ∈ 𝑇 → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))
5	4	adantr 483	. . . . 5 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))
6	5	eleq2d 2900	. . . 4 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑥 ∈ dom (𝑆 ∖ I ) ↔ 𝑥 ∈ dom (𝑄 ∖ I )))
7	4	difeq1d 4100	. . . . . 6 ⊢ (𝑄 ∈ 𝑇 → (dom (𝑆 ∖ I ) ∖ {𝑥}) = (dom (𝑄 ∖ I ) ∖ {𝑥}))
8	7	unieqd 4854	. . . . 5 ⊢ (𝑄 ∈ 𝑇 → ∪ (dom (𝑆 ∖ I ) ∖ {𝑥}) = ∪ (dom (𝑄 ∖ I ) ∖ {𝑥}))
9	8	adantr 483	. . . 4 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → ∪ (dom (𝑆 ∖ I ) ∖ {𝑥}) = ∪ (dom (𝑄 ∖ I ) ∖ {𝑥}))
10	6, 9	ifbieq1d 4492	. . 3 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → if(𝑥 ∈ dom (𝑆 ∖ I ), ∪ (dom (𝑆 ∖ I ) ∖ {𝑥}), 𝑥) = if(𝑥 ∈ dom (𝑄 ∖ I ), ∪ (dom (𝑄 ∖ I ) ∖ {𝑥}), 𝑥))
11	1, 2, 3	pmtrdifellem1 18606	. . . 4 ⊢ (𝑄 ∈ 𝑇 → 𝑆 ∈ 𝑅)
12		eldifi 4105	. . . 4 ⊢ (𝑥 ∈ (𝑁 ∖ {𝐾}) → 𝑥 ∈ 𝑁)
13		eqid 2823	. . . . 5 ⊢ (pmTrsp‘𝑁) = (pmTrsp‘𝑁)
14		eqid 2823	. . . . 5 ⊢ dom (𝑆 ∖ I ) = dom (𝑆 ∖ I )
15	13, 2, 14	pmtrffv 18589	. . . 4 ⊢ ((𝑆 ∈ 𝑅 ∧ 𝑥 ∈ 𝑁) → (𝑆‘𝑥) = if(𝑥 ∈ dom (𝑆 ∖ I ), ∪ (dom (𝑆 ∖ I ) ∖ {𝑥}), 𝑥))
16	11, 12, 15	syl2an 597	. . 3 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑆‘𝑥) = if(𝑥 ∈ dom (𝑆 ∖ I ), ∪ (dom (𝑆 ∖ I ) ∖ {𝑥}), 𝑥))
17		eqid 2823	. . . 4 ⊢ (pmTrsp‘(𝑁 ∖ {𝐾})) = (pmTrsp‘(𝑁 ∖ {𝐾}))
18		eqid 2823	. . . 4 ⊢ dom (𝑄 ∖ I ) = dom (𝑄 ∖ I )
19	17, 1, 18	pmtrffv 18589	. . 3 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑄‘𝑥) = if(𝑥 ∈ dom (𝑄 ∖ I ), ∪ (dom (𝑄 ∖ I ) ∖ {𝑥}), 𝑥))
20	10, 16, 19	3eqtr4rd 2869	. 2 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑄‘𝑥) = (𝑆‘𝑥))
21	20	ralrimiva 3184	1 ⊢ (𝑄 ∈ 𝑇 → ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑥) = (𝑆‘𝑥))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140   ∖ cdif 3935  ifcif 4469  {csn 4569  ∪ cuni 4840   I cid 5461  dom cdm 5557  ran crn 5558  ‘cfv 6357  pmTrspcpmtr 18571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-om 7583  df-1o 8104  df-2o 8105  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-pmtr 18572

This theorem is referenced by:  pmtrdifel  18610  pmtrdifwrdellem3  18613