Theorem pmtrdifellem3 19357

Description: Lemma 3 for pmtrdifel 19359. (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 19356	. . . . . 6 ⊢ (𝑄 ∈ 𝑇 → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))
5	4	adantr 480	. . . . 5 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))
6	5	eleq2d 2814	. . . 4 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑥 ∈ dom (𝑆 ∖ I ) ↔ 𝑥 ∈ dom (𝑄 ∖ I )))
7	4	difeq1d 4076	. . . . . 6 ⊢ (𝑄 ∈ 𝑇 → (dom (𝑆 ∖ I ) ∖ {𝑥}) = (dom (𝑄 ∖ I ) ∖ {𝑥}))
8	7	unieqd 4871	. . . . 5 ⊢ (𝑄 ∈ 𝑇 → ∪ (dom (𝑆 ∖ I ) ∖ {𝑥}) = ∪ (dom (𝑄 ∖ I ) ∖ {𝑥}))
9	8	adantr 480	. . . 4 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → ∪ (dom (𝑆 ∖ I ) ∖ {𝑥}) = ∪ (dom (𝑄 ∖ I ) ∖ {𝑥}))
10	6, 9	ifbieq1d 4501	. . 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 19355	. . . 4 ⊢ (𝑄 ∈ 𝑇 → 𝑆 ∈ 𝑅)
12		eldifi 4082	. . . 4 ⊢ (𝑥 ∈ (𝑁 ∖ {𝐾}) → 𝑥 ∈ 𝑁)
13		eqid 2729	. . . . 5 ⊢ (pmTrsp‘𝑁) = (pmTrsp‘𝑁)
14		eqid 2729	. . . . 5 ⊢ dom (𝑆 ∖ I ) = dom (𝑆 ∖ I )
15	13, 2, 14	pmtrffv 19338	. . . 4 ⊢ ((𝑆 ∈ 𝑅 ∧ 𝑥 ∈ 𝑁) → (𝑆‘𝑥) = if(𝑥 ∈ dom (𝑆 ∖ I ), ∪ (dom (𝑆 ∖ I ) ∖ {𝑥}), 𝑥))
16	11, 12, 15	syl2an 596	. . 3 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑆‘𝑥) = if(𝑥 ∈ dom (𝑆 ∖ I ), ∪ (dom (𝑆 ∖ I ) ∖ {𝑥}), 𝑥))
17		eqid 2729	. . . 4 ⊢ (pmTrsp‘(𝑁 ∖ {𝐾})) = (pmTrsp‘(𝑁 ∖ {𝐾}))
18		eqid 2729	. . . 4 ⊢ dom (𝑄 ∖ I ) = dom (𝑄 ∖ I )
19	17, 1, 18	pmtrffv 19338	. . 3 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑄‘𝑥) = if(𝑥 ∈ dom (𝑄 ∖ I ), ∪ (dom (𝑄 ∖ I ) ∖ {𝑥}), 𝑥))
20	10, 16, 19	3eqtr4rd 2775	. 2 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑄‘𝑥) = (𝑆‘𝑥))
21	20	ralrimiva 3121	1 ⊢ (𝑄 ∈ 𝑇 → ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑥) = (𝑆‘𝑥))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∖ cdif 3900  ifcif 4476  {csn 4577  ∪ cuni 4858   I cid 5513  dom cdm 5619  ran crn 5620  ‘cfv 6482  pmTrspcpmtr 19320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-om 7800  df-1o 8388  df-2o 8389  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-pmtr 19321

This theorem is referenced by:  pmtrdifel  19359  pmtrdifwrdellem3  19362