Theorem bnj1489

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	bnj1489.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1489.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1489.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1489.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
	bnj1489.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
	bnj1489.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
	bnj1489.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
	bnj1489.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
	bnj1489.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
	bnj1489.10	⊢ 𝑃 = ∪ 𝐻
	bnj1489.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1489.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
Assertion	bnj1489	⊢ ( 𝜒 → 𝑄 ∈ V )

Proof

Step	Hyp	Ref	Expression
1		bnj1489.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1489.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1489.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1489.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5		bnj1489.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
6		bnj1489.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
7		bnj1489.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
8		bnj1489.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
9		bnj1489.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
10		bnj1489.10	⊢ 𝑃 = ∪ 𝐻
11		bnj1489.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
12		bnj1489.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
13		bnj1364	⊢ ( 𝑅 FrSe 𝐴 → 𝑅 Se 𝐴 )
14		df-bnj13	⊢ ( 𝑅 Se 𝐴 ↔ ∀ 𝑥 ∈ 𝐴 pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V )
15	13 14	sylib	⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V )
16	6 15	bnj832	⊢ ( 𝜓 → ∀ 𝑥 ∈ 𝐴 pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V )
17	7 16	bnj835	⊢ ( 𝜒 → ∀ 𝑥 ∈ 𝐴 pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V )
18	5 7	bnj1212	⊢ ( 𝜒 → 𝑥 ∈ 𝐴 )
19	17 18	bnj1294	⊢ ( 𝜒 → pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V )
20		nfv	⊢ Ⅎ 𝑦 𝜓
21		nfv	⊢ Ⅎ 𝑦 𝑥 ∈ 𝐷
22		nfra1	⊢ Ⅎ 𝑦 ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥
23	20 21 22	nf3an	⊢ Ⅎ 𝑦 ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 )
24	7 23	nfxfr	⊢ Ⅎ 𝑦 𝜒
25	6	simplbi	⊢ ( 𝜓 → 𝑅 FrSe 𝐴 )
26	7 25	bnj835	⊢ ( 𝜒 → 𝑅 FrSe 𝐴 )
27	26	adantr	⊢ ( ( 𝜒 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → 𝑅 FrSe 𝐴 )
28	1 2 3 4 5 6 7 8	bnj1388	⊢ ( 𝜒 → ∀ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∃ 𝑓 𝜏′ )
29	28	r19.21bi	⊢ ( ( 𝜒 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ∃ 𝑓 𝜏′ )
30		nfv	⊢ Ⅎ 𝑥 𝑅 FrSe 𝐴
31		nfsbc1v	⊢ Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜏
32	8 31	nfxfr	⊢ Ⅎ 𝑥 𝜏′
33	32	nfex	⊢ Ⅎ 𝑥 ∃ 𝑓 𝜏′
34	30 33	nfan	⊢ Ⅎ 𝑥 ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 𝜏′ )
35	32	nfeuw	⊢ Ⅎ 𝑥 ∃! 𝑓 𝜏′
36	34 35	nfim	⊢ Ⅎ 𝑥 ( ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 𝜏′ ) → ∃! 𝑓 𝜏′ )
37		sneq	⊢ ( 𝑥 = 𝑦 → { 𝑥 } = { 𝑦 } )
38		bnj1318	⊢ ( 𝑥 = 𝑦 → trCl ( 𝑥 , 𝐴 , 𝑅 ) = trCl ( 𝑦 , 𝐴 , 𝑅 ) )
39	37 38	uneq12d	⊢ ( 𝑥 = 𝑦 → ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) )
40	39	eqeq2d	⊢ ( 𝑥 = 𝑦 → ( dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ↔ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
41	40	anbi2d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) )
42	1 2 3 4 8	bnj1373	⊢ ( 𝜏′ ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
43	41 42	bitr4di	⊢ ( 𝑥 = 𝑦 → ( ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ 𝜏′ ) )
44	43	exbidv	⊢ ( 𝑥 = 𝑦 → ( ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ ∃ 𝑓 𝜏′ ) )
45	44	anbi2d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) ↔ ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 𝜏′ ) ) )
46	43	eubidv	⊢ ( 𝑥 = 𝑦 → ( ∃! 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ ∃! 𝑓 𝜏′ ) )
47	45 46	imbi12d	⊢ ( 𝑥 = 𝑦 → ( ( ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) → ∃! 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 𝜏′ ) → ∃! 𝑓 𝜏′ ) ) )
48		biid	⊢ ( ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
49	1 2 3 48	bnj1321	⊢ ( ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) → ∃! 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
50	36 47 49	chvarfv	⊢ ( ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 𝜏′ ) → ∃! 𝑓 𝜏′ )
51	27 29 50	syl2anc	⊢ ( ( 𝜒 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ∃! 𝑓 𝜏′ )
52	51	ex	⊢ ( 𝜒 → ( 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) → ∃! 𝑓 𝜏′ ) )
53	24 52	ralrimi	⊢ ( 𝜒 → ∀ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∃! 𝑓 𝜏′ )
54	9	a1i	⊢ ( 𝜒 → 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ } )
55		biid	⊢ ( ( pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V ∧ ∀ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∃! 𝑓 𝜏′ ∧ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ } ) ↔ ( pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V ∧ ∀ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∃! 𝑓 𝜏′ ∧ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ } ) )
56	55	bnj1366	⊢ ( ( pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V ∧ ∀ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∃! 𝑓 𝜏′ ∧ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ } ) → 𝐻 ∈ V )
57	19 53 54 56	syl3anc	⊢ ( 𝜒 → 𝐻 ∈ V )
58	57	uniexd	⊢ ( 𝜒 → ∪ 𝐻 ∈ V )
59	10 58	eqeltrid	⊢ ( 𝜒 → 𝑃 ∈ V )
60		snex	⊢ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } ∈ V
61	60	a1i	⊢ ( 𝜒 → { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } ∈ V )
62	59 61	bnj1149	⊢ ( 𝜒 → ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } ) ∈ V )
63	12 62	eqeltrid	⊢ ( 𝜒 → 𝑄 ∈ V )