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Metamath Proof Explorer

Theorem dminss

Description: An upper bound for intersection with a domain. Theorem 40 of Suppes p. 66, who calls it "somewhat surprising". (Contributed by NM, 11-Aug-2004)

Ref Expression
Assertion dminss ( dom 𝑅 ∩ 𝐴 ) ⊆ ( ◡ 𝑅 “ ( 𝑅 “ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 19.8a ⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 𝑅 𝑦 ) → ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑥 𝑅 𝑦 ) )
2 1 ancoms ⊢ ( ( 𝑥 𝑅 𝑦 ∧ 𝑥 ∈ 𝐴 ) → ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑥 𝑅 𝑦 ) )
3 vex ⊢ 𝑦 ∈ V
4 3 elima2 ⊢ ( 𝑦 ∈ ( 𝑅 “ 𝐴 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑥 𝑅 𝑦 ) )
5 2 4 sylibr ⊢ ( ( 𝑥 𝑅 𝑦 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ ( 𝑅 “ 𝐴 ) )
6 simpl ⊢ ( ( 𝑥 𝑅 𝑦 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 𝑅 𝑦 )
7 vex ⊢ 𝑥 ∈ V
8 3 7 brcnv ⊢ ( 𝑦 ◡ 𝑅 𝑥 ↔ 𝑥 𝑅 𝑦 )
9 6 8 sylibr ⊢ ( ( 𝑥 𝑅 𝑦 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ◡ 𝑅 𝑥 )
10 5 9 jca ⊢ ( ( 𝑥 𝑅 𝑦 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑦 ∈ ( 𝑅 “ 𝐴 ) ∧ 𝑦 ◡ 𝑅 𝑥 ) )
11 10 eximi ⊢ ( ∃ 𝑦 ( 𝑥 𝑅 𝑦 ∧ 𝑥 ∈ 𝐴 ) → ∃ 𝑦 ( 𝑦 ∈ ( 𝑅 “ 𝐴 ) ∧ 𝑦 ◡ 𝑅 𝑥 ) )
12 7 eldm ⊢ ( 𝑥 ∈ dom 𝑅 ↔ ∃ 𝑦 𝑥 𝑅 𝑦 )
13 12 anbi1i ⊢ ( ( 𝑥 ∈ dom 𝑅 ∧ 𝑥 ∈ 𝐴 ) ↔ ( ∃ 𝑦 𝑥 𝑅 𝑦 ∧ 𝑥 ∈ 𝐴 ) )
14 elin ⊢ ( 𝑥 ∈ ( dom 𝑅 ∩ 𝐴 ) ↔ ( 𝑥 ∈ dom 𝑅 ∧ 𝑥 ∈ 𝐴 ) )
15 19.41v ⊢ ( ∃ 𝑦 ( 𝑥 𝑅 𝑦 ∧ 𝑥 ∈ 𝐴 ) ↔ ( ∃ 𝑦 𝑥 𝑅 𝑦 ∧ 𝑥 ∈ 𝐴 ) )
16 13 14 15 3bitr4i ⊢ ( 𝑥 ∈ ( dom 𝑅 ∩ 𝐴 ) ↔ ∃ 𝑦 ( 𝑥 𝑅 𝑦 ∧ 𝑥 ∈ 𝐴 ) )
17 7 elima2 ⊢ ( 𝑥 ∈ ( ◡ 𝑅 “ ( 𝑅 “ 𝐴 ) ) ↔ ∃ 𝑦 ( 𝑦 ∈ ( 𝑅 “ 𝐴 ) ∧ 𝑦 ◡ 𝑅 𝑥 ) )
18 11 16 17 3imtr4i ⊢ ( 𝑥 ∈ ( dom 𝑅 ∩ 𝐴 ) → 𝑥 ∈ ( ◡ 𝑅 “ ( 𝑅 “ 𝐴 ) ) )
19 18 ssriv ⊢ ( dom 𝑅 ∩ 𝐴 ) ⊆ ( ◡ 𝑅 “ ( 𝑅 “ 𝐴 ) )