Theorem noetainflem2 27710

Description: Lemma for noeta 27715. 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 5935	. . 3 ⊢ (𝑊 ↾ dom 𝑇) = ((𝑇 ∪ ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o})) ↾ dom 𝑇)
3		resundir 5954	. . 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 2760	. 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 27690	. . . . 5 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → 𝑇 ∈ No )
7		nofun 27621	. . . . 5 ⊢ (𝑇 ∈ No → Fun 𝑇)
8		funrel 6510	. . . . 5 ⊢ (Fun 𝑇 → Rel 𝑇)
9		resdm 5986	. . . . 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 5972	. . . . . . 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 4734	. . . . . . . . . 10 ⊢ {2o} ≠ ∅
14		dmxp 5879	. . . . . . . . . 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 4170	. . . . . . . 8 ⊢ (dom 𝑇 ∩ dom ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o})) = (dom 𝑇 ∩ (suc ∪ ( bday “ 𝐴) ∖ dom 𝑇))
17		disjdif 4425	. . . . . . . 8 ⊢ (dom 𝑇 ∩ (suc ∪ ( bday “ 𝐴) ∖ dom 𝑇)) = ∅
18	16, 17	eqtri 2760	. . . . . . 7 ⊢ (dom 𝑇 ∩ dom ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o})) = ∅
19	11, 18	eqtri 2760	. . . . . 6 ⊢ dom (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇) = ∅
20		relres 5965	. . . . . . 7 ⊢ Rel (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇)
21		reldm0 5878	. . . . . . 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 231	. . . . 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 4122	. . 3 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → ((𝑇 ↾ dom 𝑇) ∪ (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇)) = (𝑇 ∪ ∅))
26		un0 4347	. . 3 ⊢ (𝑇 ∪ ∅) = 𝑇
27	25, 26	eqtrdi 2788	. 2 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → ((𝑇 ↾ dom 𝑇) ∪ (((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}) ↾ dom 𝑇)) = 𝑇)
28	4, 27	eqtrid 2784	1 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → (𝑊 ↾ dom 𝑇) = 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  Vcvv 3441   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4286  ifcif 4480  {csn 4581  ⟨cop 4587  ∪ cuni 4864   class class class wbr 5099   ↦ cmpt 5180   × cxp 5623  dom cdm 5625   ↾ cres 5627   “ cima 5628  Rel wrel 5630  suc csuc 6320  ℩cio 6447  Fun wfun 6487  ‘cfv 6493  ℩crio 7316  1_oc1o 8392  2_oc2o 8393   No csur 27611   <s clts 27612   bday cbday 27613

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fo 6499  df-fv 6501  df-riota 7317  df-1o 8399  df-2o 8400  df-no 27614  df-lts 27615  df-bday 27616

This theorem is referenced by:  noetainflem3  27711  noetainflem4  27712  noetalem1  27713