Theorem noetasuplem2 28121

Description: Lemma for noeta 28130. 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 5977	. . 3 ⊢ (𝑍 ↾ dom 𝑆) = ((𝑆 ∪ ((suc ∪ (bday “ 𝐵) ∖ dom 𝑆) × {1o})) ↾ dom 𝑆)
3		resundir 5996	. . 3 ⊢ ((𝑆 ∪ ((suc ∪ (bday “ 𝐵) ∖ dom 𝑆) × {1o})) ↾ dom 𝑆) = ((𝑆 ↾ dom 𝑆) ∪ (((suc ∪ (bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆))
4		dmres 6015	. . . . . 6 ⊢ dom (((suc ∪ (bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆) = (dom 𝑆 ∩ dom ((suc ∪ (bday “ 𝐵) ∖ dom 𝑆) × {1o}))
5		1oex 8514	. . . . . . . . 9 ⊢ 1o ∈ V
6	5	snnz 4737	. . . . . . . 8 ⊢ {1o} ≠ ∅
7		dmxp 5918	. . . . . . . 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 4117	. . . . . 6 ⊢ (dom 𝑆 ∩ dom ((suc ∪ (bday “ 𝐵) ∖ dom 𝑆) × {1o})) = (dom 𝑆 ∩ (suc ∪ (bday “ 𝐵) ∖ dom 𝑆))
10		disjdif 4429	. . . . . 6 ⊢ (dom 𝑆 ∩ (suc ∪ (bday “ 𝐵) ∖ dom 𝑆)) = ∅
11	4, 9, 10	3eqtri 2790	. . . . 5 ⊢ dom (((suc ∪ (bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆) = ∅
12		relres 6007	. . . . . 6 ⊢ Rel (((suc ∪ (bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆)
13		reldm0 5917	. . . . . 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 234	. . . 4 ⊢ (((suc ∪ (bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆) = ∅
16	15	uneq2i 4167	. . 3 ⊢ ((𝑆 ↾ dom 𝑆) ∪ (((suc ∪ (bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆)) = ((𝑆 ↾ dom 𝑆) ∪ ∅)
17	2, 3, 16	3eqtri 2790	. 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 28090	. . . . . 6 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No)
20	19	3adant3 1150	. . . . 5 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑆 ∈ No)
21		nofun 28036	. . . . 5 ⊢ (𝑆 ∈ No → Fun 𝑆)
22		funrel 6576	. . . . 5 ⊢ (Fun 𝑆 → Rel 𝑆)
23		resdm 6027	. . . . 5 ⊢ (Rel 𝑆 → (𝑆 ↾ dom 𝑆) = 𝑆)
24	20, 21, 22, 23	4syl 20	. . . 4 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑆 ↾ dom 𝑆) = 𝑆)
25	24	uneq1d 4169	. . 3 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝑆 ↾ dom 𝑆) ∪ ∅) = (𝑆 ∪ ∅))
26		un0 4347	. . 3 ⊢ (𝑆 ∪ ∅) = 𝑆
27	25, 26	eqtrdi 2814	. 2 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝑆 ↾ dom 𝑆) ∪ ∅) = 𝑆)
28	17, 27	eqtrid 2810	1 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑍 ↾ dom 𝑆) = 𝑆)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  ifcif 4485  {csn 4588  ⟨cop 4838  ∪ cuni 4870   class class class wbr 5106   ↦ cmpt 5189   × cxp 5653  dom cdm 5655   ↾ cres 5657   “ cima 5658  Rel wrel 5660  suc csuc 6382  ℩cio 6510  Fun wfun 6552  ‘cfv 6558  ℩crio 7399  1_oc1o 8497  2_oc2o 8498  Nocsur 28027   <s clts 28028  bdaycbday 28029

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-fo 6564  df-fv 6566  df-riota 7400  df-1o 8504  df-2o 8505  df-no 28030  df-lts 28031  df-bday 28032

This theorem is used by:  noetalem1  28128