Theorem noetasuplem2 27874

Description: Lemma for noeta 27883. 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 5974	. . 3 ⊢ (𝑍 ↾ dom 𝑆) = ((𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) ↾ dom 𝑆)
3		resundir 5993	. . 3 ⊢ ((𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) ↾ dom 𝑆) = ((𝑆 ↾ dom 𝑆) ∪ (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆))
4		dmres 6011	. . . . . 6 ⊢ dom (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆) = (dom 𝑆 ∩ dom ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}))
5		1oex 8462	. . . . . . . . 9 ⊢ 1o ∈ V
6	5	snnz 4741	. . . . . . . 8 ⊢ {1o} ≠ ∅
7		dmxp 5919	. . . . . . . 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 4169	. . . . . 6 ⊢ (dom 𝑆 ∩ dom ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) = (dom 𝑆 ∩ (suc ∪ ( bday “ 𝐵) ∖ dom 𝑆))
10		disjdif 4432	. . . . . 6 ⊢ (dom 𝑆 ∩ (suc ∪ ( bday “ 𝐵) ∖ dom 𝑆)) = ∅
11	4, 9, 10	3eqtri 2788	. . . . 5 ⊢ dom (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆) = ∅
12		relres 6004	. . . . . 6 ⊢ Rel (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆)
13		reldm0 5918	. . . . . 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 4118	. . 3 ⊢ ((𝑆 ↾ dom 𝑆) ∪ (((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) ↾ dom 𝑆)) = ((𝑆 ↾ dom 𝑆) ∪ ∅)
17	2, 3, 16	3eqtri 2788	. 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 27843	. . . . . 6 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
20	19	3adant3 1148	. . . . 5 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑆 ∈ No )
21		nofun 27789	. . . . 5 ⊢ (𝑆 ∈ No → Fun 𝑆)
22		funrel 6553	. . . . 5 ⊢ (Fun 𝑆 → Rel 𝑆)
23		resdm 6025	. . . . 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 4120	. . 3 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝑆 ↾ dom 𝑆) ∪ ∅) = (𝑆 ∪ ∅))
26		un0 4350	. . 3 ⊢ (𝑆 ∪ ∅) = 𝑆
27	25, 26	eqtrdi 2812	. 2 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝑆 ↾ dom 𝑆) ∪ ∅) = 𝑆)
28	17, 27	eqtrid 2808	1 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑍 ↾ dom 𝑆) = 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141  {cab 2739   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  Vcvv 3453   ∖ cdif 3901   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  ifcif 4486  {csn 4588  ⟨cop 4594  ∪ cuni 4871   class class class wbr 5108   ↦ cmpt 5191   × cxp 5659  dom cdm 5661   ↾ cres 5663   “ cima 5664  Rel wrel 5666  suc csuc 6362  ℩cio 6490  Fun wfun 6530  ‘cfv 6536  ℩crio 7366  1_oc1o 8445  2_oc2o 8446   No csur 27780   <s clts 27781   bday cbday 27782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-riota 7367  df-1o 8452  df-2o 8453  df-no 27783  df-lts 27784  df-bday 27785

This theorem is referenced by:  noetalem1  27881