Theorem noetainflem2 27722

Description: Lemma for noeta 27727. 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 5933	. . 3 ⊢ (𝑊 ↾ dom 𝑇) = ((𝑇 ∪ ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o})) ↾ dom 𝑇)
3		resundir 5952	. . 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 2764	. 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 27702	. . . . 5 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → 𝑇 ∈ No )
7		nofun 27633	. . . . 5 ⊢ (𝑇 ∈ No → Fun 𝑇)
8		funrel 6505	. . . . 5 ⊢ (Fun 𝑇 → Rel 𝑇)
9		resdm 5984	. . . . 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 5970	. . . . . . 7 ⊢ dom (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = (dom 𝑇 ∩ dom ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}))
12		2oex 8410	. . . . . . . . . . 11 ⊢ 2o ∈ V
13	12	snnz 4710	. . . . . . . . . 10 ⊢ {2o} ≠ ∅
14		dmxp 5877	. . . . . . . . . 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 4148	. . . . . . . 8 ⊢ (dom 𝑇 ∩ dom ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o})) = (dom 𝑇 ∩ (suc ∪ ( bday “ 𝐴) ∖ dom 𝑇))
17		disjdif 4402	. . . . . . . 8 ⊢ (dom 𝑇 ∩ (suc ∪ ( bday “ 𝐴) ∖ dom 𝑇)) = ∅
18	16, 17	eqtri 2764	. . . . . . 7 ⊢ (dom 𝑇 ∩ dom ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o})) = ∅
19	11, 18	eqtri 2764	. . . . . 6 ⊢ dom (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = ∅
20		relres 5963	. . . . . . 7 ⊢ Rel (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇)
21		reldm0 5876	. . . . . . 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 4101	. . 3 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → ((𝑇 ↾ dom 𝑇) ∪ (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇)) = (𝑇 ∪ ∅))
26		un0 4324	. . 3 ⊢ (𝑇 ∪ ∅) = 𝑇
27	25, 26	eqtrdi 2792	. 2 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → ((𝑇 ↾ dom 𝑇) ∪ (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇)) = 𝑇)
28	4, 27	eqtrid 2788	1 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → (𝑊 ↾ dom 𝑇) = 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  {cab 2719   ≠ wne 2936  ∀wral 3055  ∃wrex 3065  Vcvv 3433   ∖ cdif 3881   ∪ cun 3882   ∩ cin 3883   ⊆ wss 3884  ∅c0 4263  ifcif 4456  {csn 4557  ⟨cop 4563  ∪ cuni 4840   class class class wbr 5074   ↦ cmpt 5155   × cxp 5618  dom cdm 5620   ↾ cres 5622   “ cima 5623  Rel wrel 5625  suc csuc 6315  ℩cio 6442  Fun wfun 6482  ‘cfv 6488  ℩crio 7315  1_oc1o 8392  2_oc2o 8393   No csur 27624   <s clts 27625   bday cbday 27626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3904  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-ord 6316  df-on 6317  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-fo 6494  df-fv 6496  df-riota 7316  df-1o 8399  df-2o 8400  df-no 27627  df-lts 27628  df-bday 27629

This theorem is referenced by:  noetainflem3  27723  noetainflem4  27724  noetalem1  27725