Theorem bnj1493

Description: Technical lemma for bnj60 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1493.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1493.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1493.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
Assertion	bnj1493	⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 ∃ 𝑓 ∈ 𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1493.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1493.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1493.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		biid	⊢ ( ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5		eqid	⊢ { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) } = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) }
6		biid	⊢ ( ( 𝑅 FrSe 𝐴 ∧ { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) } ≠ ∅ ) ↔ ( 𝑅 FrSe 𝐴 ∧ { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) } ≠ ∅ ) )
7		biid	⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) } ≠ ∅ ) ∧ 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) } ∧ ∀ 𝑦 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) } ¬ 𝑦 𝑅 𝑥 ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) } ≠ ∅ ) ∧ 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) } ∧ ∀ 𝑦 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) } ¬ 𝑦 𝑅 𝑥 ) )
8		biid	⊢ ( [ 𝑦 / 𝑥 ] ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ [ 𝑦 / 𝑥 ] ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
9		eqid	⊢ { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) [ 𝑦 / 𝑥 ] ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) } = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) [ 𝑦 / 𝑥 ] ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) }
10		eqid	⊢ ∪ { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) [ 𝑦 / 𝑥 ] ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) } = ∪ { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) [ 𝑦 / 𝑥 ] ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) }
11		eqid	⊢ ⟨ 𝑥 , ( ∪ { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) [ 𝑦 / 𝑥 ] ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) } ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ = ⟨ 𝑥 , ( ∪ { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) [ 𝑦 / 𝑥 ] ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) } ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
12		eqid	⊢ ( ∪ { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) [ 𝑦 / 𝑥 ] ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) } ∪ { ⟨ 𝑥 , ( 𝐺 ‘ ⟨ 𝑥 , ( ∪ { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) [ 𝑦 / 𝑥 ] ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) } ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ⟩ } ) = ( ∪ { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) [ 𝑦 / 𝑥 ] ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) } ∪ { ⟨ 𝑥 , ( 𝐺 ‘ ⟨ 𝑥 , ( ∪ { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) [ 𝑦 / 𝑥 ] ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) } ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ⟩ } )
13		eqid	⊢ ⟨ 𝑧 , ( ( ∪ { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) [ 𝑦 / 𝑥 ] ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) } ∪ { ⟨ 𝑥 , ( 𝐺 ‘ ⟨ 𝑥 , ( ∪ { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) [ 𝑦 / 𝑥 ] ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) } ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ⟩ } ) ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩ = ⟨ 𝑧 , ( ( ∪ { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) [ 𝑦 / 𝑥 ] ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) } ∪ { ⟨ 𝑥 , ( 𝐺 ‘ ⟨ 𝑥 , ( ∪ { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) [ 𝑦 / 𝑥 ] ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) } ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ⟩ } ) ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
14		eqid	⊢ ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
15	1 2 3 4 5 6 7 8 9 10 11 12 13 14	bnj1312	⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 ∃ 𝑓 ∈ 𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )