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Metamath Proof Explorer

Theorem exidcl

Description: Closure of the binary operation of a magma with identity. (Contributed by Jeff Madsen, 16-Jun-2011)

Ref Expression
Hypothesis exidcl.1 ⊢ 𝑋 = ran 𝐺
Assertion exidcl ( ( 𝐺 ∈ ( Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐺 𝐵 ) ∈ 𝑋 )

Proof

Step Hyp Ref Expression
1 exidcl.1 ⊢ 𝑋 = ran 𝐺
2 rngopidOLD ⊢ ( 𝐺 ∈ ( Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺 )
3 1 2 eqtrid ⊢ ( 𝐺 ∈ ( Magma ∩ ExId ) → 𝑋 = dom dom 𝐺 )
4 3 eleq2d ⊢ ( 𝐺 ∈ ( Magma ∩ ExId ) → ( 𝐴 ∈ 𝑋 ↔ 𝐴 ∈ dom dom 𝐺 ) )
5 3 eleq2d ⊢ ( 𝐺 ∈ ( Magma ∩ ExId ) → ( 𝐵 ∈ 𝑋 ↔ 𝐵 ∈ dom dom 𝐺 ) )
6 4 5 anbi12d ⊢ ( 𝐺 ∈ ( Magma ∩ ExId ) → ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ↔ ( 𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺 ) ) )
7 6 pm5.32i ⊢ ( ( 𝐺 ∈ ( Magma ∩ ExId ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) ↔ ( 𝐺 ∈ ( Magma ∩ ExId ) ∧ ( 𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺 ) ) )
8 inss1 ⊢ ( Magma ∩ ExId ) ⊆ Magma
9 8 sseli ⊢ ( 𝐺 ∈ ( Magma ∩ ExId ) → 𝐺 ∈ Magma )
10 eqid ⊢ dom dom 𝐺 = dom dom 𝐺
11 10 clmgmOLD ⊢ ( ( 𝐺 ∈ Magma ∧ 𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺 ) → ( 𝐴 𝐺 𝐵 ) ∈ dom dom 𝐺 )
12 9 11 syl3an1 ⊢ ( ( 𝐺 ∈ ( Magma ∩ ExId ) ∧ 𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺 ) → ( 𝐴 𝐺 𝐵 ) ∈ dom dom 𝐺 )
13 12 3expb ⊢ ( ( 𝐺 ∈ ( Magma ∩ ExId ) ∧ ( 𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺 ) ) → ( 𝐴 𝐺 𝐵 ) ∈ dom dom 𝐺 )
14 7 13 sylbi ⊢ ( ( 𝐺 ∈ ( Magma ∩ ExId ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐴 𝐺 𝐵 ) ∈ dom dom 𝐺 )
15 14 3impb ⊢ ( ( 𝐺 ∈ ( Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐺 𝐵 ) ∈ dom dom 𝐺 )
16 3 3ad2ant1 ⊢ ( ( 𝐺 ∈ ( Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → 𝑋 = dom dom 𝐺 )
17 15 16 eleqtrrd ⊢ ( ( 𝐺 ∈ ( Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐺 𝐵 ) ∈ 𝑋 )