Theorem pmtrdifellem3 19444

Description: Lemma 3 for pmtrdifel 19446. (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 19443	. . . . . 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 2825	. . . 4 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑥 ∈ dom (𝑆 ∖ I ) ↔ 𝑥 ∈ dom (𝑄 ∖ I )))
7	4	difeq1d 4056	. . . . . 6 ⊢ (𝑄 ∈ 𝑇 → (dom (𝑆 ∖ I ) ∖ {𝑥}) = (dom (𝑄 ∖ I ) ∖ {𝑥}))
8	7	unieqd 4851	. . . . 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 4479	. . 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 19442	. . . 4 ⊢ (𝑄 ∈ 𝑇 → 𝑆 ∈ 𝑅)
12		eldifi 4061	. . . 4 ⊢ (𝑥 ∈ (𝑁 ∖ {𝐾}) → 𝑥 ∈ 𝑁)
13		eqid 2739	. . . . 5 ⊢ (pmTrsp‘𝑁) = (pmTrsp‘𝑁)
14		eqid 2739	. . . . 5 ⊢ dom (𝑆 ∖ I ) = dom (𝑆 ∖ I )
15	13, 2, 14	pmtrffv 19425	. . . 4 ⊢ ((𝑆 ∈ 𝑅 ∧ 𝑥 ∈ 𝑁) → (𝑆‘𝑥) = if(𝑥 ∈ dom (𝑆 ∖ I ), ∪ (dom (𝑆 ∖ I ) ∖ {𝑥}), 𝑥))
16	11, 12, 15	syl2an 602	. . 3 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑆‘𝑥) = if(𝑥 ∈ dom (𝑆 ∖ I ), ∪ (dom (𝑆 ∖ I ) ∖ {𝑥}), 𝑥))
17		eqid 2739	. . . 4 ⊢ (pmTrsp‘(𝑁 ∖ {𝐾})) = (pmTrsp‘(𝑁 ∖ {𝐾}))
18		eqid 2739	. . . 4 ⊢ dom (𝑄 ∖ I ) = dom (𝑄 ∖ I )
19	17, 1, 18	pmtrffv 19425	. . 3 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑄‘𝑥) = if(𝑥 ∈ dom (𝑄 ∖ I ), ∪ (dom (𝑄 ∖ I ) ∖ {𝑥}), 𝑥))
20	10, 16, 19	3eqtr4rd 2785	. 2 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑄‘𝑥) = (𝑆‘𝑥))
21	20	ralrimiva 3131	1 ⊢ (𝑄 ∈ 𝑇 → ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑥) = (𝑆‘𝑥))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053   ∖ cdif 3880  ifcif 4454  {csn 4555  ∪ cuni 4838   I cid 5512  dom cdm 5618  ran crn 5619  ‘cfv 6485  pmTrspcpmtr 19407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-om 7807  df-1o 8395  df-2o 8396  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pmtr 19408

This theorem is referenced by:  pmtrdifel  19446  pmtrdifwrdellem3  19449