Metamath Proof Explorer


Theorem noinfdm

Description: Next, we calculate the domain of T . This is mostly to change bound variables. (Contributed by Scott Fenton, 8-Aug-2024)

Ref Expression
Hypothesis noinfdm.1 𝑇 = if ( ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 , ( ( 𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ) ∪ { ⟨ dom ( 𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ) , 1o ⟩ } ) , ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢𝐵 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥𝑢𝐵 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢𝑔 ) = 𝑥 ) ) ) )
Assertion noinfdm ( ¬ ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = { 𝑧 ∣ ∃ 𝑝𝐵 ( 𝑧 ∈ dom 𝑝 ∧ ∀ 𝑞𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) ) ) } )

Proof

Step Hyp Ref Expression
1 noinfdm.1 𝑇 = if ( ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 , ( ( 𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ) ∪ { ⟨ dom ( 𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ) , 1o ⟩ } ) , ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢𝐵 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥𝑢𝐵 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢𝑔 ) = 𝑥 ) ) ) )
2 iffalse ( ¬ ∃ 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 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥𝑢𝐵 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢𝑔 ) = 𝑥 ) ) ) ) = ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢𝐵 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥𝑢𝐵 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢𝑔 ) = 𝑥 ) ) ) )
3 1 2 syl5eq ( ¬ ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥𝑇 = ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢𝐵 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥𝑢𝐵 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢𝑔 ) = 𝑥 ) ) ) )
4 3 dmeqd ( ¬ ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = dom ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢𝐵 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥𝑢𝐵 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢𝑔 ) = 𝑥 ) ) ) )
5 iotaex ( ℩ 𝑥𝑢𝐵 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢𝑔 ) = 𝑥 ) ) ∈ V
6 eqid ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢𝐵 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥𝑢𝐵 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢𝑔 ) = 𝑥 ) ) ) = ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢𝐵 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥𝑢𝐵 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢𝑔 ) = 𝑥 ) ) )
7 5 6 dmmpti dom ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢𝐵 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥𝑢𝐵 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢𝑔 ) = 𝑥 ) ) ) = { 𝑦 ∣ ∃ 𝑢𝐵 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) }
8 eleq1w ( 𝑦 = 𝑧 → ( 𝑦 ∈ dom 𝑢𝑧 ∈ dom 𝑢 ) )
9 suceq ( 𝑦 = 𝑧 → suc 𝑦 = suc 𝑧 )
10 9 reseq2d ( 𝑦 = 𝑧 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑢 ↾ suc 𝑧 ) )
11 9 reseq2d ( 𝑦 = 𝑧 → ( 𝑣 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑧 ) )
12 10 11 eqeq12d ( 𝑦 = 𝑧 → ( ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ↔ ( 𝑢 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) )
13 12 imbi2d ( 𝑦 = 𝑧 → ( ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ↔ ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ) )
14 13 ralbidv ( 𝑦 = 𝑧 → ( ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ↔ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ) )
15 8 14 anbi12d ( 𝑦 = 𝑧 → ( ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) ↔ ( 𝑧 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ) ) )
16 15 rexbidv ( 𝑦 = 𝑧 → ( ∃ 𝑢𝐵 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) ↔ ∃ 𝑢𝐵 ( 𝑧 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ) ) )
17 dmeq ( 𝑢 = 𝑝 → dom 𝑢 = dom 𝑝 )
18 17 eleq2d ( 𝑢 = 𝑝 → ( 𝑧 ∈ dom 𝑢𝑧 ∈ dom 𝑝 ) )
19 breq1 ( 𝑢 = 𝑝 → ( 𝑢 <s 𝑣𝑝 <s 𝑣 ) )
20 19 notbid ( 𝑢 = 𝑝 → ( ¬ 𝑢 <s 𝑣 ↔ ¬ 𝑝 <s 𝑣 ) )
21 reseq1 ( 𝑢 = 𝑝 → ( 𝑢 ↾ suc 𝑧 ) = ( 𝑝 ↾ suc 𝑧 ) )
22 21 eqeq1d ( 𝑢 = 𝑝 → ( ( 𝑢 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ↔ ( 𝑝 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) )
23 20 22 imbi12d ( 𝑢 = 𝑝 → ( ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ↔ ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ) )
24 23 ralbidv ( 𝑢 = 𝑝 → ( ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ↔ ∀ 𝑣𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ) )
25 breq2 ( 𝑣 = 𝑞 → ( 𝑝 <s 𝑣𝑝 <s 𝑞 ) )
26 25 notbid ( 𝑣 = 𝑞 → ( ¬ 𝑝 <s 𝑣 ↔ ¬ 𝑝 <s 𝑞 ) )
27 reseq1 ( 𝑣 = 𝑞 → ( 𝑣 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) )
28 27 eqeq2d ( 𝑣 = 𝑞 → ( ( 𝑝 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ↔ ( 𝑝 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) ) )
29 26 28 imbi12d ( 𝑣 = 𝑞 → ( ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ↔ ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) ) ) )
30 29 cbvralvw ( ∀ 𝑣𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ↔ ∀ 𝑞𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) ) )
31 24 30 bitrdi ( 𝑢 = 𝑝 → ( ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ↔ ∀ 𝑞𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) ) ) )
32 18 31 anbi12d ( 𝑢 = 𝑝 → ( ( 𝑧 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ) ↔ ( 𝑧 ∈ dom 𝑝 ∧ ∀ 𝑞𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) ) ) ) )
33 32 cbvrexvw ( ∃ 𝑢𝐵 ( 𝑧 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ) ↔ ∃ 𝑝𝐵 ( 𝑧 ∈ dom 𝑝 ∧ ∀ 𝑞𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) ) ) )
34 16 33 bitrdi ( 𝑦 = 𝑧 → ( ∃ 𝑢𝐵 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) ↔ ∃ 𝑝𝐵 ( 𝑧 ∈ dom 𝑝 ∧ ∀ 𝑞𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) ) ) ) )
35 34 cbvabv { 𝑦 ∣ ∃ 𝑢𝐵 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } = { 𝑧 ∣ ∃ 𝑝𝐵 ( 𝑧 ∈ dom 𝑝 ∧ ∀ 𝑞𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) ) ) }
36 7 35 eqtri dom ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢𝐵 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥𝑢𝐵 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢𝑔 ) = 𝑥 ) ) ) = { 𝑧 ∣ ∃ 𝑝𝐵 ( 𝑧 ∈ dom 𝑝 ∧ ∀ 𝑞𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) ) ) }
37 4 36 eqtrdi ( ¬ ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = { 𝑧 ∣ ∃ 𝑝𝐵 ( 𝑧 ∈ dom 𝑝 ∧ ∀ 𝑞𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) ) ) } )