Theorem cdlemeiota

Description: A translation is uniquely determined by one of its values. (Contributed by NM, 18-Apr-2013)

	Ref	Expression
Hypotheses	cdlemg1c.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdlemg1c.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdlemg1c.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdlemg1c.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion	cdlemeiota	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemg1c.l	⊢ ≤ = ( le ‘ 𝐾 )
2		cdlemg1c.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3		cdlemg1c.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		cdlemg1c.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5		eqidd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) )
6		simp3	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ 𝑇 )
7	1 2 3 4	ltrnel	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) )
8	7	3com23	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) )
9	1 2 3 4	cdleme	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) ) → ∃! 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) )
10	8 9	syld3an3	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → ∃! 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) )
11		fveq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) )
12	11	eqeq1d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ↔ ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) )
13	12	riota2	⊢ ( ( 𝐹 ∈ 𝑇 ∧ ∃! 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) → ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ↔ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) = 𝐹 ) )
14	6 10 13	syl2anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ↔ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) = 𝐹 ) )
15	5 14	mpbid	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) = 𝐹 )
16	15	eqcomd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) )