Theorem noetasuplem2 27237

Description: Lemma for noeta 27246. 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 5978	. . 3 ⊢ (𝑍 ↾ dom 𝑆) = ((𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) ↾ dom 𝑆)
3		resundir 5997	. . 3 ⊢ ((𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) ↾ dom 𝑆) = ((𝑆 ↾ dom 𝑆) ∪ (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆))
4		dmres 6004	. . . . . 6 ⊢ dom (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆) = (dom 𝑆 ∩ dom ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}))
5		1oex 8476	. . . . . . . . 9 ⊢ 1o ∈ V
6	5	snnz 4781	. . . . . . . 8 ⊢ {1o} ≠ ∅
7		dmxp 5929	. . . . . . . 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 4210	. . . . . 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 2765	. . . . 5 ⊢ dom (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆) = ∅
12		relres 6011	. . . . . 6 ⊢ Rel (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆)
13		reldm0 5928	. . . . . 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 230	. . . 4 ⊢ (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆) = ∅
16	15	uneq2i 4161	. . 3 ⊢ ((𝑆 ↾ dom 𝑆) ∪ (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆)) = ((𝑆 ↾ dom 𝑆) ∪ ∅)
17	2, 3, 16	3eqtri 2765	. 2 ⊢ (𝑍 ↾ dom 𝑆) = ((𝑆 ↾ dom 𝑆) ∪ ∅)
18		noetasuplem.1	. . . . . . . 8 ⊢ 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 27206	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
20	19	3adant3 1133	. . . . . 6 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑆 ∈ No )
21		nofun 27152	. . . . . 6 ⊢ (𝑆 ∈ No → Fun 𝑆)
22	20, 21	syl 17	. . . . 5 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → Fun 𝑆)
23		funrel 6566	. . . . 5 ⊢ (Fun 𝑆 → Rel 𝑆)
24		resdm 6027	. . . . 5 ⊢ (Rel 𝑆 → (𝑆 ↾ dom 𝑆) = 𝑆)
25	22, 23, 24	3syl 18	. . . 4 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑆 ↾ dom 𝑆) = 𝑆)
26	25	uneq1d 4163	. . 3 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝑆 ↾ dom 𝑆) ∪ ∅) = (𝑆 ∪ ∅))
27		un0 4391	. . 3 ⊢ (𝑆 ∪ ∅) = 𝑆
28	26, 27	eqtrdi 2789	. 2 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝑆 ↾ dom 𝑆) ∪ ∅) = 𝑆)
29	17, 28	eqtrid 2785	1 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑍 ↾ dom 𝑆) = 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  Vcvv 3475   ∖ cdif 3946   ∪ cun 3947   ∩ cin 3948   ⊆ wss 3949  ∅c0 4323  ifcif 4529  {csn 4629  ⟨cop 4635  ∪ cuni 4909   class class class wbr 5149   ↦ cmpt 5232   × cxp 5675  dom cdm 5677   ↾ cres 5679   “ cima 5680  Rel wrel 5682  suc csuc 6367  ℩cio 6494  Fun wfun 6538  ‘cfv 6544  ℩crio 7364  1_oc1o 8459  2_oc2o 8460   No csur 27143   <s cslt 27144   bday cbday 27145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-1o 8466  df-2o 8467  df-no 27146  df-slt 27147  df-bday 27148

This theorem is referenced by:  noetalem1  27244