Metamath Proof Explorer


Theorem r1elss

Description: The range of the R1 function is transitive. Lemma 2.10 of Kunen p. 97. (Contributed by Mario Carneiro, 22-Mar-2013) (Revised by Mario Carneiro, 16-Nov-2014)

Ref Expression
Hypothesis r1elss.1 𝐴 ∈ V
Assertion r1elss ( 𝐴 ( 𝑅1 “ On ) ↔ 𝐴 ( 𝑅1 “ On ) )

Proof

Step Hyp Ref Expression
1 r1elss.1 𝐴 ∈ V
2 r1elssi ( 𝐴 ( 𝑅1 “ On ) → 𝐴 ( 𝑅1 “ On ) )
3 1 tz9.12 ( ∀ 𝑦𝐴𝑥 ∈ On 𝑦 ∈ ( 𝑅1𝑥 ) → ∃ 𝑥 ∈ On 𝐴 ∈ ( 𝑅1𝑥 ) )
4 dfss3 ( 𝐴 ( 𝑅1 “ On ) ↔ ∀ 𝑦𝐴 𝑦 ( 𝑅1 “ On ) )
5 r1fnon 𝑅1 Fn On
6 fnfun ( 𝑅1 Fn On → Fun 𝑅1 )
7 funiunfv ( Fun 𝑅1 𝑥 ∈ On ( 𝑅1𝑥 ) = ( 𝑅1 “ On ) )
8 5 6 7 mp2b 𝑥 ∈ On ( 𝑅1𝑥 ) = ( 𝑅1 “ On )
9 8 eleq2i ( 𝑦 𝑥 ∈ On ( 𝑅1𝑥 ) ↔ 𝑦 ( 𝑅1 “ On ) )
10 eliun ( 𝑦 𝑥 ∈ On ( 𝑅1𝑥 ) ↔ ∃ 𝑥 ∈ On 𝑦 ∈ ( 𝑅1𝑥 ) )
11 9 10 bitr3i ( 𝑦 ( 𝑅1 “ On ) ↔ ∃ 𝑥 ∈ On 𝑦 ∈ ( 𝑅1𝑥 ) )
12 11 ralbii ( ∀ 𝑦𝐴 𝑦 ( 𝑅1 “ On ) ↔ ∀ 𝑦𝐴𝑥 ∈ On 𝑦 ∈ ( 𝑅1𝑥 ) )
13 4 12 bitri ( 𝐴 ( 𝑅1 “ On ) ↔ ∀ 𝑦𝐴𝑥 ∈ On 𝑦 ∈ ( 𝑅1𝑥 ) )
14 8 eleq2i ( 𝐴 𝑥 ∈ On ( 𝑅1𝑥 ) ↔ 𝐴 ( 𝑅1 “ On ) )
15 eliun ( 𝐴 𝑥 ∈ On ( 𝑅1𝑥 ) ↔ ∃ 𝑥 ∈ On 𝐴 ∈ ( 𝑅1𝑥 ) )
16 14 15 bitr3i ( 𝐴 ( 𝑅1 “ On ) ↔ ∃ 𝑥 ∈ On 𝐴 ∈ ( 𝑅1𝑥 ) )
17 3 13 16 3imtr4i ( 𝐴 ( 𝑅1 “ On ) → 𝐴 ( 𝑅1 “ On ) )
18 2 17 impbii ( 𝐴 ( 𝑅1 “ On ) ↔ 𝐴 ( 𝑅1 “ On ) )