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Metamath Proof Explorer

Theorem r1ord

Description: Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of TakeutiZaring p. 77. (Contributed by NM, 8-Sep-2003) (Revised by Mario Carneiro, 16-Nov-2014)

Ref Expression
Assertion r1ord ( 𝐵 ∈ On → ( 𝐴 ∈ 𝐵 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 r1fnon ⊢ 𝑅1 Fn On
2 1 fndmi ⊢ dom 𝑅1 = On
3 2 eleq2i ⊢ ( 𝐵 ∈ dom 𝑅1 ↔ 𝐵 ∈ On )
4 r1ordg ⊢ ( 𝐵 ∈ dom 𝑅1 → ( 𝐴 ∈ 𝐵 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝐵 ) ) )
5 3 4 sylbir ⊢ ( 𝐵 ∈ On → ( 𝐴 ∈ 𝐵 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝐵 ) ) )