Theorem pmtrdifellem3 19407

Description: Lemma 3 for pmtrdifel 19409. (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 19406	. . . . . 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 2822	. . . 4 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑥 ∈ dom (𝑆 ∖ I ) ↔ 𝑥 ∈ dom (𝑄 ∖ I )))
7	4	difeq1d 4077	. . . . . 6 ⊢ (𝑄 ∈ 𝑇 → (dom (𝑆 ∖ I ) ∖ {𝑥}) = (dom (𝑄 ∖ I ) ∖ {𝑥}))
8	7	unieqd 4876	. . . . 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 4504	. . 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 19405	. . . 4 ⊢ (𝑄 ∈ 𝑇 → 𝑆 ∈ 𝑅)
12		eldifi 4083	. . . 4 ⊢ (𝑥 ∈ (𝑁 ∖ {𝐾}) → 𝑥 ∈ 𝑁)
13		eqid 2736	. . . . 5 ⊢ (pmTrsp‘𝑁) = (pmTrsp‘𝑁)
14		eqid 2736	. . . . 5 ⊢ dom (𝑆 ∖ I ) = dom (𝑆 ∖ I )
15	13, 2, 14	pmtrffv 19388	. . . 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 2736	. . . 4 ⊢ (pmTrsp‘(𝑁 ∖ {𝐾})) = (pmTrsp‘(𝑁 ∖ {𝐾}))
18		eqid 2736	. . . 4 ⊢ dom (𝑄 ∖ I ) = dom (𝑄 ∖ I )
19	17, 1, 18	pmtrffv 19388	. . 3 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑄‘𝑥) = if(𝑥 ∈ dom (𝑄 ∖ I ), ∪ (dom (𝑄 ∖ I ) ∖ {𝑥}), 𝑥))
20	10, 16, 19	3eqtr4rd 2782	. 2 ⊢ ((𝑄 ∈ 𝑇 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑄‘𝑥) = (𝑆‘𝑥))
21	20	ralrimiva 3128	1 ⊢ (𝑄 ∈ 𝑇 → ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑥) = (𝑆‘𝑥))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051   ∖ cdif 3898  ifcif 4479  {csn 4580  ∪ cuni 4863   I cid 5518  dom cdm 5624  ran crn 5625  ‘cfv 6492  pmTrspcpmtr 19370

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7809  df-1o 8397  df-2o 8398  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pmtr 19371

This theorem is referenced by:  pmtrdifel  19409  pmtrdifwrdellem3  19412