Theorem bnj1421

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	bnj1421.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1421.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1421.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1421.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
	bnj1421.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
	bnj1421.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
	bnj1421.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
	bnj1421.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
	bnj1421.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
	bnj1421.10	⊢ 𝑃 = ∪ 𝐻
	bnj1421.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1421.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
	bnj1421.13	⊢ ( 𝜒 → Fun 𝑃 )
	bnj1421.14	⊢ ( 𝜒 → dom 𝑄 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
	bnj1421.15	⊢ ( 𝜒 → dom 𝑃 = trCl ( 𝑥 , 𝐴 , 𝑅 ) )
Assertion	bnj1421	⊢ ( 𝜒 → Fun 𝑄 )

Proof

Step	Hyp	Ref	Expression
1		bnj1421.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1421.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1421.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1421.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5		bnj1421.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
6		bnj1421.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
7		bnj1421.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
8		bnj1421.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
9		bnj1421.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
10		bnj1421.10	⊢ 𝑃 = ∪ 𝐻
11		bnj1421.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
12		bnj1421.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
13		bnj1421.13	⊢ ( 𝜒 → Fun 𝑃 )
14		bnj1421.14	⊢ ( 𝜒 → dom 𝑄 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
15		bnj1421.15	⊢ ( 𝜒 → dom 𝑃 = trCl ( 𝑥 , 𝐴 , 𝑅 ) )
16		vex	⊢ 𝑥 ∈ V
17		fvex	⊢ ( 𝐺 ‘ 𝑍 ) ∈ V
18	16 17	funsn	⊢ Fun { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ }
19	13 18	jctir	⊢ ( 𝜒 → ( Fun 𝑃 ∧ Fun { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } ) )
20	17	dmsnop	⊢ dom { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } = { 𝑥 }
21	20	a1i	⊢ ( 𝜒 → dom { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } = { 𝑥 } )
22	15 21	ineq12d	⊢ ( 𝜒 → ( dom 𝑃 ∩ dom { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } ) = ( trCl ( 𝑥 , 𝐴 , 𝑅 ) ∩ { 𝑥 } ) )
23	6	simplbi	⊢ ( 𝜓 → 𝑅 FrSe 𝐴 )
24	7 23	bnj835	⊢ ( 𝜒 → 𝑅 FrSe 𝐴 )
25		biid	⊢ ( 𝑅 FrSe 𝐴 ↔ 𝑅 FrSe 𝐴 )
26		biid	⊢ ( ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ↔ ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
27		biid	⊢ ( ∀ 𝑧 ∈ 𝐴 ( 𝑧 𝑅 𝑥 → [ 𝑧 / 𝑥 ] ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ↔ ∀ 𝑧 ∈ 𝐴 ( 𝑧 𝑅 𝑥 → [ 𝑧 / 𝑥 ] ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
28		biid	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑧 𝑅 𝑥 → [ 𝑧 / 𝑥 ] ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑧 𝑅 𝑥 → [ 𝑧 / 𝑥 ] ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
29		eqid	⊢ ( pred ( 𝑥 , 𝐴 , 𝑅 ) ∪ ∪ 𝑧 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) trCl ( 𝑧 , 𝐴 , 𝑅 ) ) = ( pred ( 𝑥 , 𝐴 , 𝑅 ) ∪ ∪ 𝑧 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) trCl ( 𝑧 , 𝐴 , 𝑅 ) )
30	25 26 27 28 29	bnj1417	⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
31		disjsn	⊢ ( ( trCl ( 𝑥 , 𝐴 , 𝑅 ) ∩ { 𝑥 } ) = ∅ ↔ ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
32	31	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ( trCl ( 𝑥 , 𝐴 , 𝑅 ) ∩ { 𝑥 } ) = ∅ ↔ ∀ 𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
33	30 32	sylibr	⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 ( trCl ( 𝑥 , 𝐴 , 𝑅 ) ∩ { 𝑥 } ) = ∅ )
34	24 33	syl	⊢ ( 𝜒 → ∀ 𝑥 ∈ 𝐴 ( trCl ( 𝑥 , 𝐴 , 𝑅 ) ∩ { 𝑥 } ) = ∅ )
35	5 7	bnj1212	⊢ ( 𝜒 → 𝑥 ∈ 𝐴 )
36	34 35	bnj1294	⊢ ( 𝜒 → ( trCl ( 𝑥 , 𝐴 , 𝑅 ) ∩ { 𝑥 } ) = ∅ )
37	22 36	eqtrd	⊢ ( 𝜒 → ( dom 𝑃 ∩ dom { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } ) = ∅ )
38		funun	⊢ ( ( ( Fun 𝑃 ∧ Fun { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } ) ∧ ( dom 𝑃 ∩ dom { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } ) = ∅ ) → Fun ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } ) )
39	19 37 38	syl2anc	⊢ ( 𝜒 → Fun ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } ) )
40	12	funeqi	⊢ ( Fun 𝑄 ↔ Fun ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } ) )
41	39 40	sylibr	⊢ ( 𝜒 → Fun 𝑄 )