Metamath Proof Explorer


Theorem wdomnumr

Description: Weak dominance agrees with normal for numerable right sets. (Contributed by Stefan O'Rear, 28-Feb-2015) (Revised by Mario Carneiro, 5-May-2015)

Ref Expression
Assertion wdomnumr ( 𝐵 ∈ dom card → ( 𝐴* 𝐵𝐴𝐵 ) )

Proof

Step Hyp Ref Expression
1 brwdom ( 𝐵 ∈ dom card → ( 𝐴* 𝐵 ↔ ( 𝐴 = ∅ ∨ ∃ 𝑥 𝑥 : 𝐵onto𝐴 ) ) )
2 0domg ( 𝐵 ∈ dom card → ∅ ≼ 𝐵 )
3 breq1 ( 𝐴 = ∅ → ( 𝐴𝐵 ↔ ∅ ≼ 𝐵 ) )
4 2 3 syl5ibrcom ( 𝐵 ∈ dom card → ( 𝐴 = ∅ → 𝐴𝐵 ) )
5 fodomnum ( 𝐵 ∈ dom card → ( 𝑥 : 𝐵onto𝐴𝐴𝐵 ) )
6 5 exlimdv ( 𝐵 ∈ dom card → ( ∃ 𝑥 𝑥 : 𝐵onto𝐴𝐴𝐵 ) )
7 4 6 jaod ( 𝐵 ∈ dom card → ( ( 𝐴 = ∅ ∨ ∃ 𝑥 𝑥 : 𝐵onto𝐴 ) → 𝐴𝐵 ) )
8 1 7 sylbid ( 𝐵 ∈ dom card → ( 𝐴* 𝐵𝐴𝐵 ) )
9 domwdom ( 𝐴𝐵𝐴* 𝐵 )
10 8 9 impbid1 ( 𝐵 ∈ dom card → ( 𝐴* 𝐵𝐴𝐵 ) )