Theorem noetainflem2 28014

Description: Lemma for noeta 28019. 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 2783	. 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 27994	. . . . 5 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → 𝑇 ∈ No )
7		nofun 27925	. . . . 5 ⊢ (𝑇 ∈ No → Fun 𝑇)
8		funrel 6545	. . . . 5 ⊢ (Fun 𝑇 → Rel 𝑇)
9		resdm 6014	. . . . 5 ⊢ (Rel 𝑇 → (𝑇 ↾ dom 𝑇) = 𝑇)
10	6, 7, 8, 9	4syl 20	. . . 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 8467	. . . . . . . . . . 11 ⊢ 2o ∈ V
13	12	snnz 4737	. . . . . . . . . 10 ⊢ {2o} ≠ ∅
14		dmxp 5908	. . . . . . . . . 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 4163	. . . . . . . 8 ⊢ (dom 𝑇 ∩ dom ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o})) = (dom 𝑇 ∩ (suc ∪ ( bday “ 𝐴) ∖ dom 𝑇))
17		disjdif 4426	. . . . . . . 8 ⊢ (dom 𝑇 ∩ (suc ∪ ( bday “ 𝐴) ∖ dom 𝑇)) = ∅
18	16, 17	eqtri 2783	. . . . . . 7 ⊢ (dom 𝑇 ∩ dom ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o})) = ∅
19	11, 18	eqtri 2783	. . . . . 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 5907	. . . . . . 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 234	. . . . 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 4116	. . 3 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → ((𝑇 ↾ dom 𝑇) ∪ (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇)) = (𝑇 ∪ ∅))
26		un0 4344	. . 3 ⊢ (𝑇 ∪ ∅) = 𝑇
27	25, 26	eqtrdi 2811	. 2 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → ((𝑇 ↾ dom 𝑇) ∪ (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇)) = 𝑇)
28	4, 27	eqtrid 2807	1 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → (𝑊 ↾ dom 𝑇) = 𝑇)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  ifcif 4482  {csn 4584  ⟨cop 4590  ∪ cuni 4867   class class class wbr 5103   ↦ cmpt 5186   × cxp 5646  dom cdm 5648   ↾ cres 5650   “ cima 5651  Rel wrel 5653  suc csuc 6354  ℩cio 6482  Fun wfun 6522  ‘cfv 6528  ℩crio 7365  1_oc1o 8448  2_oc2o 8449   No csur 27916   <s clts 27917   bday cbday 27918

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-ord 6355  df-on 6356  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-fo 6534  df-fv 6536  df-riota 7366  df-1o 8455  df-2o 8456  df-no 27919  df-lts 27920  df-bday 27921

This theorem is used by:  noetainflem3  28015  noetainflem4  28016  noetalem1  28017