Metamath Proof Explorer


Theorem r1elssi

Description: The range of the R1 function is transitive. Lemma 2.10 of Kunen p. 97. One direction of r1elss that doesn't need A to be a set. (Contributed by Mario Carneiro, 22-Mar-2013) (Revised by Mario Carneiro, 16-Nov-2014)

Ref Expression
Assertion r1elssi ( 𝐴 ( 𝑅1 “ On ) → 𝐴 ( 𝑅1 “ On ) )

Proof

Step Hyp Ref Expression
1 triun ( ∀ 𝑥 ∈ On Tr ( 𝑅1𝑥 ) → Tr 𝑥 ∈ On ( 𝑅1𝑥 ) )
2 r1tr Tr ( 𝑅1𝑥 )
3 2 a1i ( 𝑥 ∈ On → Tr ( 𝑅1𝑥 ) )
4 1 3 mprg Tr 𝑥 ∈ On ( 𝑅1𝑥 )
5 r1funlim ( Fun 𝑅1 ∧ Lim dom 𝑅1 )
6 5 simpli Fun 𝑅1
7 funiunfv ( Fun 𝑅1 𝑥 ∈ On ( 𝑅1𝑥 ) = ( 𝑅1 “ On ) )
8 6 7 ax-mp 𝑥 ∈ On ( 𝑅1𝑥 ) = ( 𝑅1 “ On )
9 treq ( 𝑥 ∈ On ( 𝑅1𝑥 ) = ( 𝑅1 “ On ) → ( Tr 𝑥 ∈ On ( 𝑅1𝑥 ) ↔ Tr ( 𝑅1 “ On ) ) )
10 8 9 ax-mp ( Tr 𝑥 ∈ On ( 𝑅1𝑥 ) ↔ Tr ( 𝑅1 “ On ) )
11 4 10 mpbi Tr ( 𝑅1 “ On )
12 trss ( Tr ( 𝑅1 “ On ) → ( 𝐴 ( 𝑅1 “ On ) → 𝐴 ( 𝑅1 “ On ) ) )
13 11 12 ax-mp ( 𝐴 ( 𝑅1 “ On ) → 𝐴 ( 𝑅1 “ On ) )