Theorem noetasuplem2 27779

Description: Lemma for noeta 27788. The restriction of 𝑍 to dom 𝑆 is 𝑆. (Contributed by Scott Fenton, 9-Aug-2024.)

Hypotheses

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

Assertion

Ref	Expression
noetasuplem2	⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑍 ↾ dom 𝑆) = 𝑆)

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

Allowed substitution hints:   𝐵(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑆(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑍(𝑥,𝑦,𝑣,𝑢,𝑔)

Proof of Theorem noetasuplem2

Step	Hyp	Ref	Expression
1		noetasuplem.2	. . . 4 ⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}))
2	1	reseq1i 5993	. . 3 ⊢ (𝑍 ↾ dom 𝑆) = ((𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) ↾ dom 𝑆)
3		resundir 6012	. . 3 ⊢ ((𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) ↾ dom 𝑆) = ((𝑆 ↾ dom 𝑆) ∪ (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆))
4		dmres 6030	. . . . . 6 ⊢ dom (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆) = (dom 𝑆 ∩ dom ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}))
5		1oex 8516	. . . . . . . . 9 ⊢ 1o ∈ V
6	5	snnz 4776	. . . . . . . 8 ⊢ {1o} ≠ ∅
7		dmxp 5939	. . . . . . . 8 ⊢ ({1o} ≠ ∅ → dom ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) = (suc ∪ ( bday “ 𝐵) ∖ dom 𝑆))
8	6, 7	ax-mp 5	. . . . . . 7 ⊢ dom ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) = (suc ∪ ( bday “ 𝐵) ∖ dom 𝑆)
9	8	ineq2i 4217	. . . . . 6 ⊢ (dom 𝑆 ∩ dom ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) = (dom 𝑆 ∩ (suc ∪ ( bday “ 𝐵) ∖ dom 𝑆))
10		disjdif 4472	. . . . . 6 ⊢ (dom 𝑆 ∩ (suc ∪ ( bday “ 𝐵) ∖ dom 𝑆)) = ∅
11	4, 9, 10	3eqtri 2769	. . . . 5 ⊢ dom (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆) = ∅
12		relres 6023	. . . . . 6 ⊢ Rel (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆)
13		reldm0 5938	. . . . . 6 ⊢ (Rel (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆) → ((((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆) = ∅ ↔ dom (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆) = ∅))
14	12, 13	ax-mp 5	. . . . 5 ⊢ ((((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆) = ∅ ↔ dom (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆) = ∅)
15	11, 14	mpbir 231	. . . 4 ⊢ (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆) = ∅
16	15	uneq2i 4165	. . 3 ⊢ ((𝑆 ↾ dom 𝑆) ∪ (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆)) = ((𝑆 ↾ dom 𝑆) ∪ ∅)
17	2, 3, 16	3eqtri 2769	. 2 ⊢ (𝑍 ↾ dom 𝑆) = ((𝑆 ↾ dom 𝑆) ∪ ∅)
18		noetasuplem.1	. . . . . . 7 ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
19	18	nosupno 27748	. . . . . 6 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
20	19	3adant3 1133	. . . . 5 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑆 ∈ No )
21		nofun 27694	. . . . 5 ⊢ (𝑆 ∈ No → Fun 𝑆)
22		funrel 6583	. . . . 5 ⊢ (Fun 𝑆 → Rel 𝑆)
23		resdm 6044	. . . . 5 ⊢ (Rel 𝑆 → (𝑆 ↾ dom 𝑆) = 𝑆)
24	20, 21, 22, 23	4syl 19	. . . 4 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑆 ↾ dom 𝑆) = 𝑆)
25	24	uneq1d 4167	. . 3 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝑆 ↾ dom 𝑆) ∪ ∅) = (𝑆 ∪ ∅))
26		un0 4394	. . 3 ⊢ (𝑆 ∪ ∅) = 𝑆
27	25, 26	eqtrdi 2793	. 2 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝑆 ↾ dom 𝑆) ∪ ∅) = 𝑆)
28	17, 27	eqtrid 2789	1 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑍 ↾ dom 𝑆) = 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  {cab 2714   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  Vcvv 3480   ∖ cdif 3948   ∪ cun 3949   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  ifcif 4525  {csn 4626  ⟨cop 4632  ∪ cuni 4907   class class class wbr 5143   ↦ cmpt 5225   × cxp 5683  dom cdm 5685   ↾ cres 5687   “ cima 5688  Rel wrel 5690  suc csuc 6386  ℩cio 6512  Fun wfun 6555  ‘cfv 6561  ℩crio 7387  1_oc1o 8499  2_oc2o 8500   No csur 27684   <s cslt 27685   bday cbday 27686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-riota 7388  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689

This theorem is referenced by:  noetalem1  27786