Metamath Proof Explorer

Theorem wfrdmcl

Description: Given F = wrecs ( R , A , X ) /\ X e. dom F , then its predecessor class is a subset of dom F . (Contributed by Scott Fenton, 21-Apr-2011)

Ref Expression
Hypothesis wfrlem6.1 ⊢ 𝐹 = wrecs ( 𝑅 , 𝐴 , 𝐺 )
Assertion wfrdmcl ( 𝑋 ∈ dom 𝐹 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ dom 𝐹 )

Proof

Step Hyp Ref Expression
1 wfrlem6.1 ⊢ 𝐹 = wrecs ( 𝑅 , 𝐴 , 𝐺 )
2 df-wrecs ⊢ wrecs ( 𝑅 , 𝐴 , 𝐺 ) = ∪ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
3 1 2 eqtri ⊢ 𝐹 = ∪ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
4 3 dmeqi ⊢ dom 𝐹 = dom ∪ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
5 dmuni ⊢ dom ∪ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } = ∪ 𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } dom 𝑔
6 4 5 eqtri ⊢ dom 𝐹 = ∪ 𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } dom 𝑔
7 6 eleq2i ⊢ ( 𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ ∪ 𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } dom 𝑔 )
8 eliun ⊢ ( 𝑋 ∈ ∪ 𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } dom 𝑔 ↔ ∃ 𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } 𝑋 ∈ dom 𝑔 )
9 7 8 bitri ⊢ ( 𝑋 ∈ dom 𝐹 ↔ ∃ 𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } 𝑋 ∈ dom 𝑔 )
10 eqid ⊢ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
11 10 wfrlem1 ⊢ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } = { 𝑔 ∣ ∃ 𝑧 ( 𝑔 Fn 𝑧 ∧ ( 𝑧 ⊆ 𝐴 ∧ ∀ 𝑤 ∈ 𝑧 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑧 ) ∧ ∀ 𝑤 ∈ 𝑧 ( 𝑔 ‘ 𝑤 ) = ( 𝐺 ‘ ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) }
12 11 abeq2i ⊢ ( 𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ↔ ∃ 𝑧 ( 𝑔 Fn 𝑧 ∧ ( 𝑧 ⊆ 𝐴 ∧ ∀ 𝑤 ∈ 𝑧 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑧 ) ∧ ∀ 𝑤 ∈ 𝑧 ( 𝑔 ‘ 𝑤 ) = ( 𝐺 ‘ ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) )
13 predeq3 ⊢ ( 𝑤 = 𝑋 → Pred ( 𝑅 , 𝐴 , 𝑤 ) = Pred ( 𝑅 , 𝐴 , 𝑋 ) )
14 13 sseq1d ⊢ ( 𝑤 = 𝑋 → ( Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑧 ↔ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝑧 ) )
15 14 rspccv ⊢ ( ∀ 𝑤 ∈ 𝑧 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑧 → ( 𝑋 ∈ 𝑧 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝑧 ) )
16 15 adantl ⊢ ( ( 𝑔 Fn 𝑧 ∧ ∀ 𝑤 ∈ 𝑧 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑧 ) → ( 𝑋 ∈ 𝑧 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝑧 ) )
17 fndm ⊢ ( 𝑔 Fn 𝑧 → dom 𝑔 = 𝑧 )
18 17 eleq2d ⊢ ( 𝑔 Fn 𝑧 → ( 𝑋 ∈ dom 𝑔 ↔ 𝑋 ∈ 𝑧 ) )
19 17 sseq2d ⊢ ( 𝑔 Fn 𝑧 → ( Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ dom 𝑔 ↔ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝑧 ) )
20 18 19 imbi12d ⊢ ( 𝑔 Fn 𝑧 → ( ( 𝑋 ∈ dom 𝑔 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ dom 𝑔 ) ↔ ( 𝑋 ∈ 𝑧 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝑧 ) ) )
21 20 adantr ⊢ ( ( 𝑔 Fn 𝑧 ∧ ∀ 𝑤 ∈ 𝑧 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑧 ) → ( ( 𝑋 ∈ dom 𝑔 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ dom 𝑔 ) ↔ ( 𝑋 ∈ 𝑧 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝑧 ) ) )
22 16 21 mpbird ⊢ ( ( 𝑔 Fn 𝑧 ∧ ∀ 𝑤 ∈ 𝑧 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑧 ) → ( 𝑋 ∈ dom 𝑔 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ dom 𝑔 ) )
23 22 adantrl ⊢ ( ( 𝑔 Fn 𝑧 ∧ ( 𝑧 ⊆ 𝐴 ∧ ∀ 𝑤 ∈ 𝑧 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑧 ) ) → ( 𝑋 ∈ dom 𝑔 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ dom 𝑔 ) )
24 23 3adant3 ⊢ ( ( 𝑔 Fn 𝑧 ∧ ( 𝑧 ⊆ 𝐴 ∧ ∀ 𝑤 ∈ 𝑧 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑧 ) ∧ ∀ 𝑤 ∈ 𝑧 ( 𝑔 ‘ 𝑤 ) = ( 𝐺 ‘ ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) → ( 𝑋 ∈ dom 𝑔 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ dom 𝑔 ) )
25 24 exlimiv ⊢ ( ∃ 𝑧 ( 𝑔 Fn 𝑧 ∧ ( 𝑧 ⊆ 𝐴 ∧ ∀ 𝑤 ∈ 𝑧 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑧 ) ∧ ∀ 𝑤 ∈ 𝑧 ( 𝑔 ‘ 𝑤 ) = ( 𝐺 ‘ ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) → ( 𝑋 ∈ dom 𝑔 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ dom 𝑔 ) )
26 12 25 sylbi ⊢ ( 𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } → ( 𝑋 ∈ dom 𝑔 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ dom 𝑔 ) )
27 26 reximia ⊢ ( ∃ 𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } 𝑋 ∈ dom 𝑔 → ∃ 𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ dom 𝑔 )
28 ssiun ⊢ ( ∃ 𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ dom 𝑔 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ ∪ 𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } dom 𝑔 )
29 27 28 syl ⊢ ( ∃ 𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } 𝑋 ∈ dom 𝑔 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ ∪ 𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } dom 𝑔 )
30 9 29 sylbi ⊢ ( 𝑋 ∈ dom 𝐹 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ ∪ 𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } dom 𝑔 )
31 30 6 sseqtrrdi ⊢ ( 𝑋 ∈ dom 𝐹 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ dom 𝐹 )