Theorem noetasuplem2 27717

Description: Lemma for noeta 27726. 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 5932	. . 3 ⊢ (𝑍 ↾ dom 𝑆) = ((𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) ↾ dom 𝑆)
3		resundir 5951	. . 3 ⊢ ((𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) ↾ dom 𝑆) = ((𝑆 ↾ dom 𝑆) ∪ (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆))
4		dmres 5969	. . . . . 6 ⊢ dom (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆) = (dom 𝑆 ∩ dom ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}))
5		1oex 8406	. . . . . . . . 9 ⊢ 1o ∈ V
6	5	snnz 4721	. . . . . . . 8 ⊢ {1o} ≠ ∅
7		dmxp 5876	. . . . . . . 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 4158	. . . . . 6 ⊢ (dom 𝑆 ∩ dom ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) = (dom 𝑆 ∩ (suc ∪ ( bday “ 𝐵) ∖ dom 𝑆))
10		disjdif 4413	. . . . . 6 ⊢ (dom 𝑆 ∩ (suc ∪ ( bday “ 𝐵) ∖ dom 𝑆)) = ∅
11	4, 9, 10	3eqtri 2764	. . . . 5 ⊢ dom (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆) = ∅
12		relres 5962	. . . . . 6 ⊢ Rel (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆)
13		reldm0 5875	. . . . . 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 4106	. . 3 ⊢ ((𝑆 ↾ dom 𝑆) ∪ (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆)) = ((𝑆 ↾ dom 𝑆) ∪ ∅)
17	2, 3, 16	3eqtri 2764	. 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 27686	. . . . . 6 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
20	19	3adant3 1133	. . . . 5 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑆 ∈ No )
21		nofun 27632	. . . . 5 ⊢ (𝑆 ∈ No → Fun 𝑆)
22		funrel 6507	. . . . 5 ⊢ (Fun 𝑆 → Rel 𝑆)
23		resdm 5983	. . . . 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 4108	. . 3 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝑆 ↾ dom 𝑆) ∪ ∅) = (𝑆 ∪ ∅))
26		un0 4335	. . 3 ⊢ (𝑆 ∪ ∅) = 𝑆
27	25, 26	eqtrdi 2788	. 2 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝑆 ↾ dom 𝑆) ∪ ∅) = 𝑆)
28	17, 27	eqtrid 2784	1 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑍 ↾ dom 𝑆) = 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ∖ cdif 3887   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  ifcif 4467  {csn 4568  ⟨cop 4574  ∪ cuni 4851   class class class wbr 5086   ↦ cmpt 5167   × cxp 5620  dom cdm 5622   ↾ cres 5624   “ cima 5625  Rel wrel 5627  suc csuc 6317  ℩cio 6444  Fun wfun 6484  ‘cfv 6490  ℩crio 7314  1_oc1o 8389  2_oc2o 8390   No csur 27622   <s clts 27623   bday cbday 27624

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

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 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-ord 6318  df-on 6319  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fo 6496  df-fv 6498  df-riota 7315  df-1o 8396  df-2o 8397  df-no 27625  df-lts 27626  df-bday 27627

This theorem is referenced by:  noetalem1  27724