Theorem bnj1384

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	bnj1384.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1384.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1384.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1384.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
	bnj1384.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
	bnj1384.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
	bnj1384.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
	bnj1384.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
	bnj1384.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
	bnj1384.10	⊢ 𝑃 = ∪ 𝐻
Assertion	bnj1384	⊢ ( 𝑅 FrSe 𝐴 → Fun 𝑃 )

Proof

Step	Hyp	Ref	Expression
1		bnj1384.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1384.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1384.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1384.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5		bnj1384.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
6		bnj1384.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
7		bnj1384.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
8		bnj1384.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
9		bnj1384.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
10		bnj1384.10	⊢ 𝑃 = ∪ 𝐻
11	1 2 3 4 8	bnj1373	⊢ ( 𝜏′ ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
12	1 2 3 4 5 6 7 8 9 10 11	bnj1371	⊢ ( 𝑓 ∈ 𝐻 → Fun 𝑓 )
13	12	rgen	⊢ ∀ 𝑓 ∈ 𝐻 Fun 𝑓
14		id	⊢ ( 𝑅 FrSe 𝐴 → 𝑅 FrSe 𝐴 )
15	1 2 3 4 5 6 7 8 9	bnj1374	⊢ ( 𝑓 ∈ 𝐻 → 𝑓 ∈ 𝐶 )
16		nfab1	⊢ Ⅎ 𝑓 { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
17	9 16	nfcxfr	⊢ Ⅎ 𝑓 𝐻
18	17	nfcri	⊢ Ⅎ 𝑓 𝑔 ∈ 𝐻
19		nfab1	⊢ Ⅎ 𝑓 { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
20	3 19	nfcxfr	⊢ Ⅎ 𝑓 𝐶
21	20	nfcri	⊢ Ⅎ 𝑓 𝑔 ∈ 𝐶
22	18 21	nfim	⊢ Ⅎ 𝑓 ( 𝑔 ∈ 𝐻 → 𝑔 ∈ 𝐶 )
23		eleq1w	⊢ ( 𝑓 = 𝑔 → ( 𝑓 ∈ 𝐻 ↔ 𝑔 ∈ 𝐻 ) )
24		eleq1w	⊢ ( 𝑓 = 𝑔 → ( 𝑓 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶 ) )
25	23 24	imbi12d	⊢ ( 𝑓 = 𝑔 → ( ( 𝑓 ∈ 𝐻 → 𝑓 ∈ 𝐶 ) ↔ ( 𝑔 ∈ 𝐻 → 𝑔 ∈ 𝐶 ) ) )
26	22 25 15	chvarfv	⊢ ( 𝑔 ∈ 𝐻 → 𝑔 ∈ 𝐶 )
27		eqid	⊢ ( dom 𝑓 ∩ dom 𝑔 ) = ( dom 𝑓 ∩ dom 𝑔 )
28	1 2 3 27	bnj1326	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑓 ∈ 𝐶 ∧ 𝑔 ∈ 𝐶 ) → ( 𝑓 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) = ( 𝑔 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) )
29	14 15 26 28	syl3an	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑓 ∈ 𝐻 ∧ 𝑔 ∈ 𝐻 ) → ( 𝑓 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) = ( 𝑔 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) )
30	29	3expib	⊢ ( 𝑅 FrSe 𝐴 → ( ( 𝑓 ∈ 𝐻 ∧ 𝑔 ∈ 𝐻 ) → ( 𝑓 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) = ( 𝑔 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) ) )
31	30	ralrimivv	⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑓 ∈ 𝐻 ∀ 𝑔 ∈ 𝐻 ( 𝑓 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) = ( 𝑔 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) )
32		biid	⊢ ( ∀ 𝑓 ∈ 𝐻 Fun 𝑓 ↔ ∀ 𝑓 ∈ 𝐻 Fun 𝑓 )
33		biid	⊢ ( ( ∀ 𝑓 ∈ 𝐻 Fun 𝑓 ∧ ∀ 𝑓 ∈ 𝐻 ∀ 𝑔 ∈ 𝐻 ( 𝑓 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) = ( 𝑔 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) ) ↔ ( ∀ 𝑓 ∈ 𝐻 Fun 𝑓 ∧ ∀ 𝑓 ∈ 𝐻 ∀ 𝑔 ∈ 𝐻 ( 𝑓 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) = ( 𝑔 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) ) )
34	9	bnj1317	⊢ ( 𝑧 ∈ 𝐻 → ∀ 𝑓 𝑧 ∈ 𝐻 )
35	32 27 33 34	bnj1386	⊢ ( ( ∀ 𝑓 ∈ 𝐻 Fun 𝑓 ∧ ∀ 𝑓 ∈ 𝐻 ∀ 𝑔 ∈ 𝐻 ( 𝑓 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) = ( 𝑔 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) ) → Fun ∪ 𝐻 )
36	13 31 35	sylancr	⊢ ( 𝑅 FrSe 𝐴 → Fun ∪ 𝐻 )
37	10	funeqi	⊢ ( Fun 𝑃 ↔ Fun ∪ 𝐻 )
38	36 37	sylibr	⊢ ( 𝑅 FrSe 𝐴 → Fun 𝑃 )