Metamath Proof Explorer


Theorem rdglim2

Description: The value of the recursive definition generator at a limit ordinal, in terms of the union of all smaller values. (Contributed by NM, 23-Apr-1995)

Ref Expression
Assertion rdglim2 B C Lim B rec F A B = y | x B y = rec F A x

Proof

Step Hyp Ref Expression
1 rdglim B C Lim B rec F A B = rec F A B
2 dfima3 rec F A B = y | x x B x y rec F A
3 df-rex x B y = rec F A x x x B y = rec F A x
4 limord Lim B Ord B
5 ordelord Ord B x B Ord x
6 5 ex Ord B x B Ord x
7 vex x V
8 7 elon x On Ord x
9 6 8 syl6ibr Ord B x B x On
10 4 9 syl Lim B x B x On
11 eqcom y = rec F A x rec F A x = y
12 rdgfnon rec F A Fn On
13 fnopfvb rec F A Fn On x On rec F A x = y x y rec F A
14 12 13 mpan x On rec F A x = y x y rec F A
15 11 14 syl5bb x On y = rec F A x x y rec F A
16 10 15 syl6 Lim B x B y = rec F A x x y rec F A
17 16 pm5.32d Lim B x B y = rec F A x x B x y rec F A
18 17 exbidv Lim B x x B y = rec F A x x x B x y rec F A
19 3 18 bitr2id Lim B x x B x y rec F A x B y = rec F A x
20 19 abbidv Lim B y | x x B x y rec F A = y | x B y = rec F A x
21 2 20 eqtrid Lim B rec F A B = y | x B y = rec F A x
22 21 unieqd Lim B rec F A B = y | x B y = rec F A x
23 22 adantl B C Lim B rec F A B = y | x B y = rec F A x
24 1 23 eqtrd B C Lim B rec F A B = y | x B y = rec F A x