Theorem bnj1423

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	bnj1423.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1423.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1423.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1423.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
	bnj1423.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
	bnj1423.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
	bnj1423.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
	bnj1423.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
	bnj1423.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
	bnj1423.10	⊢ 𝑃 = ∪ 𝐻
	bnj1423.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1423.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
	bnj1423.13	⊢ 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
	bnj1423.14	⊢ 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
	bnj1423.15	⊢ ( 𝜒 → 𝑃 Fn trCl ( 𝑥 , 𝐴 , 𝑅 ) )
	bnj1423.16	⊢ ( 𝜒 → 𝑄 Fn ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
Assertion	bnj1423	⊢ ( 𝜒 → ∀ 𝑧 ∈ 𝐸 ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1423.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1423.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1423.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1423.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5		bnj1423.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
6		bnj1423.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
7		bnj1423.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
8		bnj1423.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
9		bnj1423.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
10		bnj1423.10	⊢ 𝑃 = ∪ 𝐻
11		bnj1423.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
12		bnj1423.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
13		bnj1423.13	⊢ 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
14		bnj1423.14	⊢ 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
15		bnj1423.15	⊢ ( 𝜒 → 𝑃 Fn trCl ( 𝑥 , 𝐴 , 𝑅 ) )
16		bnj1423.16	⊢ ( 𝜒 → 𝑄 Fn ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
17		biid	⊢ ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ↔ ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) )
18		biid	⊢ ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ { 𝑥 } ) ↔ ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ { 𝑥 } ) )
19	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18	bnj1442	⊢ ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ { 𝑥 } ) → ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 ) )
20		biid	⊢ ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ↔ ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
21		biid	⊢ ( ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓 ) ↔ ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓 ) )
22		biid	⊢ ( ( ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓 ) ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) ↔ ( ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓 ) ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
23		biid	⊢ ( ( ( ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓 ) ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) ∧ 𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) ↔ ( ( ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓 ) ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) ∧ 𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) )
24		eqid	⊢ ⟨ 𝑧 , ( 𝑓 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩ = ⟨ 𝑧 , ( 𝑓 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
25	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20 21 22 23 24	bnj1450	⊢ ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 ) )
26	14	bnj1424	⊢ ( 𝑧 ∈ 𝐸 → ( 𝑧 ∈ { 𝑥 } ∨ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
27	26	adantl	⊢ ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) → ( 𝑧 ∈ { 𝑥 } ∨ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
28	19 25 27	mpjaodan	⊢ ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) → ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 ) )
29	28	ralrimiva	⊢ ( 𝜒 → ∀ 𝑧 ∈ 𝐸 ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 ) )