Theorem pmtrdifellem3 19345

Description: Lemma 3 for pmtrdifel 19347. (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 19344	. . . . . 6 ⊢ (𝑄 ∈ 𝑇 → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))
5	4	adantr 481	. . . . 5 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))
6	5	eleq2d 2819	. . . 4 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑥 ∈ dom (𝑆 ∖ I ) ↔ 𝑥 ∈ dom (𝑄 ∖ I )))
7	4	difeq1d 4121	. . . . . 6 ⊢ (𝑄 ∈ 𝑇 → (dom (𝑆 ∖ I ) ∖ {𝑥}) = (dom (𝑄 ∖ I ) ∖ {𝑥}))
8	7	unieqd 4922	. . . . 5 ⊢ (𝑄 ∈ 𝑇 → ∪ (dom (𝑆 ∖ I ) ∖ {𝑥}) = ∪ (dom (𝑄 ∖ I ) ∖ {𝑥}))
9	8	adantr 481	. . . 4 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → ∪ (dom (𝑆 ∖ I ) ∖ {𝑥}) = ∪ (dom (𝑄 ∖ I ) ∖ {𝑥}))
10	6, 9	ifbieq1d 4552	. . 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 19343	. . . 4 ⊢ (𝑄 ∈ 𝑇 → 𝑆 ∈ 𝑅)
12		eldifi 4126	. . . 4 ⊢ (𝑥 ∈ (𝑁 ∖ {𝐾}) → 𝑥 ∈ 𝑁)
13		eqid 2732	. . . . 5 ⊢ (pmTrsp‘𝑁) = (pmTrsp‘𝑁)
14		eqid 2732	. . . . 5 ⊢ dom (𝑆 ∖ I ) = dom (𝑆 ∖ I )
15	13, 2, 14	pmtrffv 19326	. . . 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 2732	. . . 4 ⊢ (pmTrsp‘(𝑁 ∖ {𝐾})) = (pmTrsp‘(𝑁 ∖ {𝐾}))
18		eqid 2732	. . . 4 ⊢ dom (𝑄 ∖ I ) = dom (𝑄 ∖ I )
19	17, 1, 18	pmtrffv 19326	. . 3 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑄‘𝑥) = if(𝑥 ∈ dom (𝑄 ∖ I ), ∪ (dom (𝑄 ∖ I ) ∖ {𝑥}), 𝑥))
20	10, 16, 19	3eqtr4rd 2783	. 2 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑄‘𝑥) = (𝑆‘𝑥))
21	20	ralrimiva 3146	1 ⊢ (𝑄 ∈ 𝑇 → ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑥) = (𝑆‘𝑥))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061   ∖ cdif 3945  ifcif 4528  {csn 4628  ∪ cuni 4908   I cid 5573  dom cdm 5676  ran crn 5677  ‘cfv 6543  pmTrspcpmtr 19308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7855  df-1o 8465  df-2o 8466  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-pmtr 19309

This theorem is referenced by:  pmtrdifel  19347  pmtrdifwrdellem3  19350