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Metamath Proof Explorer

Theorem frrlem3

Description: Lemma for well-founded recursion. An acceptable function's domain is a subset of A . (Contributed by Paul Chapman, 21-Apr-2012)

Ref Expression
Hypothesis frrlem1.1 ⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
Assertion frrlem3 ( 𝑔 ∈ 𝐵 → dom 𝑔 ⊆ 𝐴 )

Proof

Step Hyp Ref Expression
1 frrlem1.1 ⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
2 1 frrlem1 ⊢ 𝐵 = { 𝑔 ∣ ∃ 𝑧 ( 𝑔 Fn 𝑧 ∧ ( 𝑧 ⊆ 𝐴 ∧ ∀ 𝑤 ∈ 𝑧 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑧 ) ∧ ∀ 𝑤 ∈ 𝑧 ( 𝑔 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) }
3 2 eqabri ⊢ ( 𝑔 ∈ 𝐵 ↔ ∃ 𝑧 ( 𝑔 Fn 𝑧 ∧ ( 𝑧 ⊆ 𝐴 ∧ ∀ 𝑤 ∈ 𝑧 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑧 ) ∧ ∀ 𝑤 ∈ 𝑧 ( 𝑔 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) )
4 fndm ⊢ ( 𝑔 Fn 𝑧 → dom 𝑔 = 𝑧 )
5 4 sseq1d ⊢ ( 𝑔 Fn 𝑧 → ( dom 𝑔 ⊆ 𝐴 ↔ 𝑧 ⊆ 𝐴 ) )
6 5 biimpar ⊢ ( ( 𝑔 Fn 𝑧 ∧ 𝑧 ⊆ 𝐴 ) → dom 𝑔 ⊆ 𝐴 )
7 6 adantrr ⊢ ( ( 𝑔 Fn 𝑧 ∧ ( 𝑧 ⊆ 𝐴 ∧ ∀ 𝑤 ∈ 𝑧 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑧 ) ) → dom 𝑔 ⊆ 𝐴 )
8 7 3adant3 ⊢ ( ( 𝑔 Fn 𝑧 ∧ ( 𝑧 ⊆ 𝐴 ∧ ∀ 𝑤 ∈ 𝑧 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑧 ) ∧ ∀ 𝑤 ∈ 𝑧 ( 𝑔 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) → dom 𝑔 ⊆ 𝐴 )
9 8 exlimiv ⊢ ( ∃ 𝑧 ( 𝑔 Fn 𝑧 ∧ ( 𝑧 ⊆ 𝐴 ∧ ∀ 𝑤 ∈ 𝑧 Pred ( 𝑅 , 𝐴 , 𝑤 ) ⊆ 𝑧 ) ∧ ∀ 𝑤 ∈ 𝑧 ( 𝑔 ‘ 𝑤 ) = ( 𝑤 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑤 ) ) ) ) → dom 𝑔 ⊆ 𝐴 )
10 3 9 sylbi ⊢ ( 𝑔 ∈ 𝐵 → dom 𝑔 ⊆ 𝐴 )