Theorem dfwrecs2

Description: TODO: Replace df-wrecs with this definition, and shorten theorems using wrecs with it. (Contributed by BJ, 27-Oct-2024)

	Ref	Expression
Assertion	dfwrecs2	⊢ wrecs ( 𝑅 , 𝐴 , 𝐹 ) = frecs ( 𝑅 , 𝐴 , ( 𝐹 ∘ 2nd ) )

Proof

Step	Hyp	Ref	Expression
1		vex	⊢ 𝑦 ∈ V
2	1	a1i	⊢ ( ⊤ → 𝑦 ∈ V )
3		vex	⊢ 𝑓 ∈ V
4	3	resex	⊢ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ∈ V
5	4	a1i	⊢ ( ⊤ → ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ∈ V )
6	2 5	opco2	⊢ ( ⊤ → ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) = ( 𝐹 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) )
7	6	mptru	⊢ ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) = ( 𝐹 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) )
8	7	eqeq2i	⊢ ( ( 𝑓 ‘ 𝑦 ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ↔ ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) )
9	8	ralbii	⊢ ( ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ↔ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) )
10	9	3anbi3i	⊢ ( ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ↔ ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) )
11	10	exbii	⊢ ( ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ↔ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) )
12	11	abbii	⊢ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
13	12	unieqi	⊢ ∪ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } = ∪ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
14		df-frecs	⊢ frecs ( 𝑅 , 𝐴 , ( 𝐹 ∘ 2nd ) ) = ∪ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
15		df-wrecs	⊢ wrecs ( 𝑅 , 𝐴 , 𝐹 ) = ∪ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
16	13 14 15	3eqtr4ri	⊢ wrecs ( 𝑅 , 𝐴 , 𝐹 ) = frecs ( 𝑅 , 𝐴 , ( 𝐹 ∘ 2nd ) )