Theorem noetainflem2 27804

Description: Lemma for noeta 27809. The restriction of 𝑊 to the domain of 𝑇 is 𝑇. (Contributed by Scott Fenton, 9-Aug-2024.)

Hypotheses

Ref	Expression
noetainflem.1	⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
noetainflem.2	⊢ 𝑊 = (𝑇 ∪ ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}))

Assertion

Ref	Expression
noetainflem2	⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → (𝑊 ↾ dom 𝑇) = 𝑇)

Distinct variable group:   𝐵,𝑔,𝑢,𝑣,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑇(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑊(𝑥,𝑦,𝑣,𝑢,𝑔)

Proof of Theorem noetainflem2

Step	Hyp	Ref	Expression
1		noetainflem.2	. . . 4 ⊢ 𝑊 = (𝑇 ∪ ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}))
2	1	reseq1i 5963	. . 3 ⊢ (𝑊 ↾ dom 𝑇) = ((𝑇 ∪ ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o})) ↾ dom 𝑇)
3		resundir 5982	. . 3 ⊢ ((𝑇 ∪ ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o})) ↾ dom 𝑇) = ((𝑇 ↾ dom 𝑇) ∪ (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇))
4	2, 3	eqtri 2787	. 2 ⊢ (𝑊 ↾ dom 𝑇) = ((𝑇 ↾ dom 𝑇) ∪ (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇))
5		noetainflem.1	. . . . . 6 ⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
6	5	noinfno 27784	. . . . 5 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → 𝑇 ∈ No )
7		nofun 27715	. . . . 5 ⊢ (𝑇 ∈ No → Fun 𝑇)
8		funrel 6540	. . . . 5 ⊢ (Fun 𝑇 → Rel 𝑇)
9		resdm 6014	. . . . 5 ⊢ (Rel 𝑇 → (𝑇 ↾ dom 𝑇) = 𝑇)
10	6, 7, 8, 9	4syl 19	. . . 4 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → (𝑇 ↾ dom 𝑇) = 𝑇)
11		dmres 6000	. . . . . . 7 ⊢ dom (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = (dom 𝑇 ∩ dom ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}))
12		2oex 8451	. . . . . . . . . . 11 ⊢ 2o ∈ V
13	12	snnz 4737	. . . . . . . . . 10 ⊢ {2o} ≠ ∅
14		dmxp 5907	. . . . . . . . . 10 ⊢ ({2o} ≠ ∅ → dom ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) = (suc ∪ ( bday “ 𝐴) ∖ dom 𝑇))
15	13, 14	ax-mp 5	. . . . . . . . 9 ⊢ dom ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) = (suc ∪ ( bday “ 𝐴) ∖ dom 𝑇)
16	15	ineq2i 4171	. . . . . . . 8 ⊢ (dom 𝑇 ∩ dom ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o})) = (dom 𝑇 ∩ (suc ∪ ( bday “ 𝐴) ∖ dom 𝑇))
17		disjdif 4428	. . . . . . . 8 ⊢ (dom 𝑇 ∩ (suc ∪ ( bday “ 𝐴) ∖ dom 𝑇)) = ∅
18	16, 17	eqtri 2787	. . . . . . 7 ⊢ (dom 𝑇 ∩ dom ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o})) = ∅
19	11, 18	eqtri 2787	. . . . . 6 ⊢ dom (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = ∅
20		relres 5993	. . . . . . 7 ⊢ Rel (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇)
21		reldm0 5906	. . . . . . 7 ⊢ (Rel (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) → ((((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = ∅ ↔ dom (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = ∅))
22	20, 21	ax-mp 5	. . . . . 6 ⊢ ((((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = ∅ ↔ dom (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = ∅)
23	19, 22	mpbir 233	. . . . 5 ⊢ (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = ∅
24	23	a1i 11	. . . 4 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = ∅)
25	10, 24	uneq12d 4124	. . 3 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → ((𝑇 ↾ dom 𝑇) ∪ (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇)) = (𝑇 ∪ ∅))
26		un0 4350	. . 3 ⊢ (𝑇 ∪ ∅) = 𝑇
27	25, 26	eqtrdi 2815	. 2 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → ((𝑇 ↾ dom 𝑇) ∪ (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇)) = 𝑇)
28	4, 27	eqtrid 2811	1 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → (𝑊 ↾ dom 𝑇) = 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  {cab 2742   ≠ wne 2959  ∀wral 3078  ∃wrex 3088  Vcvv 3456   ∖ cdif 3903   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  ifcif 4482  {csn 4584  ⟨cop 4590  ∪ cuni 4867   class class class wbr 5102   ↦ cmpt 5183   × cxp 5647  dom cdm 5649   ↾ cres 5651   “ cima 5652  Rel wrel 5654  suc csuc 6350  ℩cio 6477  Fun wfun 6517  ‘cfv 6523  ℩crio 7354  1_oc1o 8432  2_oc2o 8433   No csur 27706   <s clts 27707   bday cbday 27708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ord 6351  df-on 6352  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-fo 6529  df-fv 6531  df-riota 7355  df-1o 8439  df-2o 8440  df-no 27709  df-lts 27710  df-bday 27711

This theorem is referenced by:  noetainflem3  27805  noetainflem4  27806  noetalem1  27807