Metamath Proof Explorer


Theorem frr1

Description: Law of general well-founded recursion, part one. This theorem and the following two drop the partial order requirement from fpr1 , fpr2 , and fpr3 , which requires using the axiom of infinity (Contributed by Scott Fenton, 11-Sep-2023)

Ref Expression
Hypothesis frr.1 𝐹 = frecs ( 𝑅 , 𝐴 , 𝐺 )
Assertion frr1 ( ( 𝑅 Fr 𝐴𝑅 Se 𝐴 ) → 𝐹 Fn 𝐴 )

Proof

Step Hyp Ref Expression
1 frr.1 𝐹 = frecs ( 𝑅 , 𝐴 , 𝐺 )
2 eqid { 𝑎 ∣ ∃ 𝑏 ( 𝑎 Fn 𝑏 ∧ ( 𝑏𝐴 ∧ ∀ 𝑐𝑏 Pred ( 𝑅 , 𝐴 , 𝑐 ) ⊆ 𝑏 ) ∧ ∀ 𝑐𝑏 ( 𝑎𝑐 ) = ( 𝑐 𝐺 ( 𝑎 ↾ Pred ( 𝑅 , 𝐴 , 𝑐 ) ) ) ) } = { 𝑎 ∣ ∃ 𝑏 ( 𝑎 Fn 𝑏 ∧ ( 𝑏𝐴 ∧ ∀ 𝑐𝑏 Pred ( 𝑅 , 𝐴 , 𝑐 ) ⊆ 𝑏 ) ∧ ∀ 𝑐𝑏 ( 𝑎𝑐 ) = ( 𝑐 𝐺 ( 𝑎 ↾ Pred ( 𝑅 , 𝐴 , 𝑐 ) ) ) ) }
3 2 frrlem1 { 𝑎 ∣ ∃ 𝑏 ( 𝑎 Fn 𝑏 ∧ ( 𝑏𝐴 ∧ ∀ 𝑐𝑏 Pred ( 𝑅 , 𝐴 , 𝑐 ) ⊆ 𝑏 ) ∧ ∀ 𝑐𝑏 ( 𝑎𝑐 ) = ( 𝑐 𝐺 ( 𝑎 ↾ Pred ( 𝑅 , 𝐴 , 𝑐 ) ) ) ) } = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝑦 𝐺 ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
4 3 1 frrlem15 ( ( ( 𝑅 Fr 𝐴𝑅 Se 𝐴 ) ∧ ( 𝑔 ∈ { 𝑎 ∣ ∃ 𝑏 ( 𝑎 Fn 𝑏 ∧ ( 𝑏𝐴 ∧ ∀ 𝑐𝑏 Pred ( 𝑅 , 𝐴 , 𝑐 ) ⊆ 𝑏 ) ∧ ∀ 𝑐𝑏 ( 𝑎𝑐 ) = ( 𝑐 𝐺 ( 𝑎 ↾ Pred ( 𝑅 , 𝐴 , 𝑐 ) ) ) ) } ∧ ∈ { 𝑎 ∣ ∃ 𝑏 ( 𝑎 Fn 𝑏 ∧ ( 𝑏𝐴 ∧ ∀ 𝑐𝑏 Pred ( 𝑅 , 𝐴 , 𝑐 ) ⊆ 𝑏 ) ∧ ∀ 𝑐𝑏 ( 𝑎𝑐 ) = ( 𝑐 𝐺 ( 𝑎 ↾ Pred ( 𝑅 , 𝐴 , 𝑐 ) ) ) ) } ) ) → ( ( 𝑥 𝑔 𝑢𝑥 𝑣 ) → 𝑢 = 𝑣 ) )
5 3 1 4 frrlem9 ( ( 𝑅 Fr 𝐴𝑅 Se 𝐴 ) → Fun 𝐹 )
6 eqid ( ( 𝐹 ↾ TrPred ( 𝑅 , 𝐴 , 𝑧 ) ) ∪ { ⟨ 𝑧 , ( 𝑧 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ⟩ } ) = ( ( 𝐹 ↾ TrPred ( 𝑅 , 𝐴 , 𝑧 ) ) ∪ { ⟨ 𝑧 , ( 𝑧 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ⟩ } )
7 simpl ( ( 𝑅 Fr 𝐴𝑅 Se 𝐴 ) → 𝑅 Fr 𝐴 )
8 setlikespec ( ( 𝑧𝐴𝑅 Se 𝐴 ) → Pred ( 𝑅 , 𝐴 , 𝑧 ) ∈ V )
9 8 ancoms ( ( 𝑅 Se 𝐴𝑧𝐴 ) → Pred ( 𝑅 , 𝐴 , 𝑧 ) ∈ V )
10 9 adantll ( ( ( 𝑅 Fr 𝐴𝑅 Se 𝐴 ) ∧ 𝑧𝐴 ) → Pred ( 𝑅 , 𝐴 , 𝑧 ) ∈ V )
11 trpredpred ( Pred ( 𝑅 , 𝐴 , 𝑧 ) ∈ V → Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ TrPred ( 𝑅 , 𝐴 , 𝑧 ) )
12 10 11 syl ( ( ( 𝑅 Fr 𝐴𝑅 Se 𝐴 ) ∧ 𝑧𝐴 ) → Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ TrPred ( 𝑅 , 𝐴 , 𝑧 ) )
13 frrlem16 ( ( ( 𝑅 Fr 𝐴𝑅 Se 𝐴 ) ∧ 𝑧𝐴 ) → ∀ 𝑎 ∈ TrPred ( 𝑅 , 𝐴 , 𝑧 ) Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ TrPred ( 𝑅 , 𝐴 , 𝑧 ) )
14 trpredex TrPred ( 𝑅 , 𝐴 , 𝑧 ) ∈ V
15 14 a1i ( ( ( 𝑅 Fr 𝐴𝑅 Se 𝐴 ) ∧ 𝑧𝐴 ) → TrPred ( 𝑅 , 𝐴 , 𝑧 ) ∈ V )
16 trpredss ( Pred ( 𝑅 , 𝐴 , 𝑧 ) ∈ V → TrPred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ 𝐴 )
17 10 16 syl ( ( ( 𝑅 Fr 𝐴𝑅 Se 𝐴 ) ∧ 𝑧𝐴 ) → TrPred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ 𝐴 )
18 difss ( 𝐴 ∖ dom 𝐹 ) ⊆ 𝐴
19 frmin ( ( ( 𝑅 Fr 𝐴𝑅 Se 𝐴 ) ∧ ( ( 𝐴 ∖ dom 𝐹 ) ⊆ 𝐴 ∧ ( 𝐴 ∖ dom 𝐹 ) ≠ ∅ ) ) → ∃ 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ )
20 18 19 mpanr1 ( ( ( 𝑅 Fr 𝐴𝑅 Se 𝐴 ) ∧ ( 𝐴 ∖ dom 𝐹 ) ≠ ∅ ) → ∃ 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ )
21 3 1 4 6 7 12 13 15 17 20 frrlem14 ( ( 𝑅 Fr 𝐴𝑅 Se 𝐴 ) → dom 𝐹 = 𝐴 )
22 df-fn ( 𝐹 Fn 𝐴 ↔ ( Fun 𝐹 ∧ dom 𝐹 = 𝐴 ) )
23 5 21 22 sylanbrc ( ( 𝑅 Fr 𝐴𝑅 Se 𝐴 ) → 𝐹 Fn 𝐴 )