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Theorem noinfdm 33487
 Description: Next, we calculate the domain of 𝑇. This is mostly to change bound variables. (Contributed by Scott Fenton, 8-Aug-2024.)
Hypothesis
Ref Expression
noinfdm.1 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
Assertion
Ref Expression
noinfdm (¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = {𝑧 ∣ ∃𝑝𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞𝐵𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))})
Distinct variable groups:   𝐵,𝑔   𝐵,𝑝,𝑞,𝑢,𝑣,𝑦,𝑧   𝑢,𝑔,𝑣,𝑦
Allowed substitution hints:   𝐵(𝑥)   𝑇(𝑥,𝑦,𝑧,𝑣,𝑢,𝑔,𝑞,𝑝)

Proof of Theorem noinfdm
StepHypRef Expression
1 noinfdm.1 . . . 4 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
2 iffalse 4429 . . . 4 (¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))) = (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
31, 2syl5eq 2805 . . 3 (¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥𝑇 = (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
43dmeqd 5745 . 2 (¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = dom (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
5 iotaex 6315 . . . 4 (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) ∈ V
6 eqid 2758 . . . 4 (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) = (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))
75, 6dmmpti 6475 . . 3 dom (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) = {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}
8 eleq1w 2834 . . . . . . 7 (𝑦 = 𝑧 → (𝑦 ∈ dom 𝑢𝑧 ∈ dom 𝑢))
9 suceq 6234 . . . . . . . . . . 11 (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
109reseq2d 5823 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑧))
119reseq2d 5823 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑧))
1210, 11eqeq12d 2774 . . . . . . . . 9 (𝑦 = 𝑧 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧)))
1312imbi2d 344 . . . . . . . 8 (𝑦 = 𝑧 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧))))
1413ralbidv 3126 . . . . . . 7 (𝑦 = 𝑧 → (∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧))))
158, 14anbi12d 633 . . . . . 6 (𝑦 = 𝑧 → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ (𝑧 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧)))))
1615rexbidv 3221 . . . . 5 (𝑦 = 𝑧 → (∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑢𝐵 (𝑧 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧)))))
17 dmeq 5743 . . . . . . . 8 (𝑢 = 𝑝 → dom 𝑢 = dom 𝑝)
1817eleq2d 2837 . . . . . . 7 (𝑢 = 𝑝 → (𝑧 ∈ dom 𝑢𝑧 ∈ dom 𝑝))
19 breq1 5035 . . . . . . . . . . 11 (𝑢 = 𝑝 → (𝑢 <s 𝑣𝑝 <s 𝑣))
2019notbid 321 . . . . . . . . . 10 (𝑢 = 𝑝 → (¬ 𝑢 <s 𝑣 ↔ ¬ 𝑝 <s 𝑣))
21 reseq1 5817 . . . . . . . . . . 11 (𝑢 = 𝑝 → (𝑢 ↾ suc 𝑧) = (𝑝 ↾ suc 𝑧))
2221eqeq1d 2760 . . . . . . . . . 10 (𝑢 = 𝑝 → ((𝑢 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧) ↔ (𝑝 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧)))
2320, 22imbi12d 348 . . . . . . . . 9 (𝑢 = 𝑝 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧)) ↔ (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧))))
2423ralbidv 3126 . . . . . . . 8 (𝑢 = 𝑝 → (∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧)) ↔ ∀𝑣𝐵𝑝 <s 𝑣 → (𝑝 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧))))
25 breq2 5036 . . . . . . . . . . 11 (𝑣 = 𝑞 → (𝑝 <s 𝑣𝑝 <s 𝑞))
2625notbid 321 . . . . . . . . . 10 (𝑣 = 𝑞 → (¬ 𝑝 <s 𝑣 ↔ ¬ 𝑝 <s 𝑞))
27 reseq1 5817 . . . . . . . . . . 11 (𝑣 = 𝑞 → (𝑣 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧))
2827eqeq2d 2769 . . . . . . . . . 10 (𝑣 = 𝑞 → ((𝑝 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧) ↔ (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))
2926, 28imbi12d 348 . . . . . . . . 9 (𝑣 = 𝑞 → ((¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧)) ↔ (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧))))
3029cbvralvw 3361 . . . . . . . 8 (∀𝑣𝐵𝑝 <s 𝑣 → (𝑝 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧)) ↔ ∀𝑞𝐵𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))
3124, 30bitrdi 290 . . . . . . 7 (𝑢 = 𝑝 → (∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧)) ↔ ∀𝑞𝐵𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧))))
3218, 31anbi12d 633 . . . . . 6 (𝑢 = 𝑝 → ((𝑧 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧))) ↔ (𝑧 ∈ dom 𝑝 ∧ ∀𝑞𝐵𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))))
3332cbvrexvw 3362 . . . . 5 (∃𝑢𝐵 (𝑧 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑧) = (𝑣 ↾ suc 𝑧))) ↔ ∃𝑝𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞𝐵𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧))))
3416, 33bitrdi 290 . . . 4 (𝑦 = 𝑧 → (∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑝𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞𝐵𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))))
3534cbvabv 2826 . . 3 {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} = {𝑧 ∣ ∃𝑝𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞𝐵𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))}
367, 35eqtri 2781 . 2 dom (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) = {𝑧 ∣ ∃𝑝𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞𝐵𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))}
374, 36eqtrdi 2809 1 (¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = {𝑧 ∣ ∃𝑝𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞𝐵𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {cab 2735  ∀wral 3070  ∃wrex 3071   ∪ cun 3856  ifcif 4420  {csn 4522  ⟨cop 4528   class class class wbr 5032   ↦ cmpt 5112  dom cdm 5524   ↾ cres 5526  suc csuc 6171  ℩cio 6292  ‘cfv 6335  ℩crio 7107  1oc1o 8105
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