Theorem noetainflem2 33530

Description: Lemma for noeta 33535. 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 5823	. . 3 ⊢ (𝑊 ↾ dom 𝑇) = ((𝑇 ∪ ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o})) ↾ dom 𝑇)
3		resundir 5842	. . 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 2781	. 2 ⊢ (𝑊 ↾ dom 𝑇) = ((𝑇 ↾ dom 𝑇) ∪ (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇))
5		noetainflem.1	. . . . . . 7 ⊢ 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 33510	. . . . . 6 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → 𝑇 ∈ No )
7		nofun 33441	. . . . . 6 ⊢ (𝑇 ∈ No → Fun 𝑇)
8	6, 7	syl 17	. . . . 5 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → Fun 𝑇)
9		funrel 6356	. . . . 5 ⊢ (Fun 𝑇 → Rel 𝑇)
10		resdm 5872	. . . . 5 ⊢ (Rel 𝑇 → (𝑇 ↾ dom 𝑇) = 𝑇)
11	8, 9, 10	3syl 18	. . . 4 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → (𝑇 ↾ dom 𝑇) = 𝑇)
12		dmres 5849	. . . . . . 7 ⊢ dom (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = (dom 𝑇 ∩ dom ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}))
13		2oex 8127	. . . . . . . . . . 11 ⊢ 2o ∈ V
14	13	snnz 4672	. . . . . . . . . 10 ⊢ {2o} ≠ ∅
15		dmxp 5774	. . . . . . . . . 10 ⊢ ({2o} ≠ ∅ → dom ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) = (suc ∪ ( bday “ 𝐴) ∖ dom 𝑇))
16	14, 15	ax-mp 5	. . . . . . . . 9 ⊢ dom ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) = (suc ∪ ( bday “ 𝐴) ∖ dom 𝑇)
17	16	ineq2i 4116	. . . . . . . 8 ⊢ (dom 𝑇 ∩ dom ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o})) = (dom 𝑇 ∩ (suc ∪ ( bday “ 𝐴) ∖ dom 𝑇))
18		disjdif 4371	. . . . . . . 8 ⊢ (dom 𝑇 ∩ (suc ∪ ( bday “ 𝐴) ∖ dom 𝑇)) = ∅
19	17, 18	eqtri 2781	. . . . . . 7 ⊢ (dom 𝑇 ∩ dom ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o})) = ∅
20	12, 19	eqtri 2781	. . . . . 6 ⊢ dom (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = ∅
21		relres 5856	. . . . . . 7 ⊢ Rel (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇)
22		reldm0 5773	. . . . . . 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) = ∅))
23	21, 22	ax-mp 5	. . . . . 6 ⊢ ((((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = ∅ ↔ dom (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = ∅)
24	20, 23	mpbir 234	. . . . 5 ⊢ (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = ∅
25	24	a1i 11	. . . 4 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = ∅)
26	11, 25	uneq12d 4071	. . 3 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → ((𝑇 ↾ dom 𝑇) ∪ (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇)) = (𝑇 ∪ ∅))
27		un0 4289	. . 3 ⊢ (𝑇 ∪ ∅) = 𝑇
28	26, 27	eqtrdi 2809	. 2 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → ((𝑇 ↾ dom 𝑇) ∪ (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇)) = 𝑇)
29	4, 28	syl5eq 2805	1 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → (𝑊 ↾ dom 𝑇) = 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {cab 2735   ≠ wne 2951  ∀wral 3070  ∃wrex 3071  Vcvv 3409   ∖ cdif 3857   ∪ cun 3858   ∩ cin 3859   ⊆ wss 3860  ∅c0 4227  ifcif 4423  {csn 4525  ⟨cop 4531  ∪ cuni 4801   class class class wbr 5035   ↦ cmpt 5115   × cxp 5525  dom cdm 5527   ↾ cres 5529   “ cima 5530  Rel wrel 5532  suc csuc 6175  ℩cio 6296  Fun wfun 6333  ‘cfv 6339  ℩crio 7112  1_oc1o 8110  2_oc2o 8111   No csur 33432   <s cslt 33433   bday cbday 33434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pr 5301  ax-un 7464

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-ord 6176  df-on 6177  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7113  df-1o 8117  df-2o 8118  df-no 33435  df-slt 33436  df-bday 33437

This theorem is referenced by:  noetainflem3  33531  noetainflem4  33532  noetalem1  33533