This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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 ⊢ ( ∀ 𝑦 ∈ 𝐴 ∃ 𝑥 ∈ On 𝑦 ∈ ( 𝑅1 ‘ 𝑥 ) → ∃ 𝑥 ∈ On 𝐴 ∈ ( 𝑅1 ‘ 𝑥 ) )
4 dfss3 ⊢ ( 𝐴 ⊆ ∪ ( 𝑅1 “ On ) ↔ ∀ 𝑦 ∈ 𝐴 𝑦 ∈ ∪ ( 𝑅1 “ On ) )
5 r1fnon ⊢ 𝑅1 Fn On
6 fnfun ⊢ ( 𝑅1 Fn On → Fun 𝑅1 )
7 funiunfv ⊢ ( Fun 𝑅1 → ∪ 𝑥 ∈ On ( 𝑅1 ‘ 𝑥 ) = ∪ ( 𝑅1 “ On ) )
8 5 6 7 mp2b ⊢ ∪ 𝑥 ∈ On ( 𝑅1 ‘ 𝑥 ) = ∪ ( 𝑅1 “ On )
9 8 eleq2i ⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ On ( 𝑅1 ‘ 𝑥 ) ↔ 𝑦 ∈ ∪ ( 𝑅1 “ On ) )
10 eliun ⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ On ( 𝑅1 ‘ 𝑥 ) ↔ ∃ 𝑥 ∈ On 𝑦 ∈ ( 𝑅1 ‘ 𝑥 ) )
11 9 10 bitr3i ⊢ ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ↔ ∃ 𝑥 ∈ On 𝑦 ∈ ( 𝑅1 ‘ 𝑥 ) )
12 11 ralbii ⊢ ( ∀ 𝑦 ∈ 𝐴 𝑦 ∈ ∪ ( 𝑅1 “ On ) ↔ ∀ 𝑦 ∈ 𝐴 ∃ 𝑥 ∈ On 𝑦 ∈ ( 𝑅1 ‘ 𝑥 ) )
13 4 12 bitri ⊢ ( 𝐴 ⊆ ∪ ( 𝑅1 “ On ) ↔ ∀ 𝑦 ∈ 𝐴 ∃ 𝑥 ∈ On 𝑦 ∈ ( 𝑅1 ‘ 𝑥 ) )
14 8 eleq2i ⊢ ( 𝐴 ∈ ∪ 𝑥 ∈ On ( 𝑅1 ‘ 𝑥 ) ↔ 𝐴 ∈ ∪ ( 𝑅1 “ On ) )
15 eliun ⊢ ( 𝐴 ∈ ∪ 𝑥 ∈ On ( 𝑅1 ‘ 𝑥 ) ↔ ∃ 𝑥 ∈ On 𝐴 ∈ ( 𝑅1 ‘ 𝑥 ) )
16 14 15 bitr3i ⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ↔ ∃ 𝑥 ∈ On 𝐴 ∈ ( 𝑅1 ‘ 𝑥 ) )
17 3 13 16 3imtr4i ⊢ ( 𝐴 ⊆ ∪ ( 𝑅1 “ On ) → 𝐴 ∈ ∪ ( 𝑅1 “ On ) )
18 2 17 impbii ⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ↔ 𝐴 ⊆ ∪ ( 𝑅1 “ On ) )