bnj1312

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	bnj1312.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1312.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1312.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1312.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
	bnj1312.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
	bnj1312.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
	bnj1312.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
	bnj1312.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
	bnj1312.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
	bnj1312.10	⊢ 𝑃 = ∪ 𝐻
	bnj1312.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1312.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
	bnj1312.13	⊢ 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
	bnj1312.14	⊢ 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
Assertion	bnj1312	⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 ∃ 𝑓 ∈ 𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1312.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1312.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1312.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1312.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5		bnj1312.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
6		bnj1312.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
7		bnj1312.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
8		bnj1312.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
9		bnj1312.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
10		bnj1312.10	⊢ 𝑃 = ∪ 𝐻
11		bnj1312.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
12		bnj1312.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
13		bnj1312.13	⊢ 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
14		bnj1312.14	⊢ 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
15	6	simplbi	⊢ ( 𝜓 → 𝑅 FrSe 𝐴 )
16	5	ssrab3	⊢ 𝐷 ⊆ 𝐴
17	16	a1i	⊢ ( 𝜓 → 𝐷 ⊆ 𝐴 )
18	6	simprbi	⊢ ( 𝜓 → 𝐷 ≠ ∅ )
19	5	bnj1230	⊢ ( 𝑤 ∈ 𝐷 → ∀ 𝑥 𝑤 ∈ 𝐷 )
20	19	bnj1228	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ 𝐷 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐷 ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 )
21	15 17 18 20	syl3anc	⊢ ( 𝜓 → ∃ 𝑥 ∈ 𝐷 ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 )
22		nfv	⊢ Ⅎ 𝑥 𝑅 FrSe 𝐴
23	19	nfcii	⊢ Ⅎ 𝑥 𝐷
24		nfcv	⊢ Ⅎ 𝑥 ∅
25	23 24	nfne	⊢ Ⅎ 𝑥 𝐷 ≠ ∅
26	22 25	nfan	⊢ Ⅎ 𝑥 ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ )
27	6 26	nfxfr	⊢ Ⅎ 𝑥 𝜓
28	27	nf5ri	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
29	21 7 28	bnj1521	⊢ ( 𝜓 → ∃ 𝑥 𝜒 )
30	7	simp2bi	⊢ ( 𝜒 → 𝑥 ∈ 𝐷 )
31	5	bnj1538	⊢ ( 𝑥 ∈ 𝐷 → ¬ ∃ 𝑓 𝜏 )
32	1 2 3 4 5 6 7 8 9 10 11 12	bnj1489	⊢ ( 𝜒 → 𝑄 ∈ V )
33	7 15	bnj835	⊢ ( 𝜒 → 𝑅 FrSe 𝐴 )
34	1 2 3 4 5 6 7 8 9 10	bnj1384	⊢ ( 𝑅 FrSe 𝐴 → Fun 𝑃 )
35	33 34	syl	⊢ ( 𝜒 → Fun 𝑃 )
36	1 2 3 4 5 6 7 8 9 10	bnj1415	⊢ ( 𝜒 → dom 𝑃 = trCl ( 𝑥 , 𝐴 , 𝑅 ) )
37	35 36	bnj1422	⊢ ( 𝜒 → 𝑃 Fn trCl ( 𝑥 , 𝐴 , 𝑅 ) )
38	1 2 3 4 5 6 7 8 9 10 11 12 36	bnj1416	⊢ ( 𝜒 → dom 𝑄 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
39	1 2 3 4 5 6 7 8 9 10 11 12 35 38 36	bnj1421	⊢ ( 𝜒 → Fun 𝑄 )
40	39 38	bnj1422	⊢ ( 𝜒 → 𝑄 Fn ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
41	1 2 3 4 5 6 7 8 9 10 11 12 13 14 37 40	bnj1423	⊢ ( 𝜒 → ∀ 𝑧 ∈ 𝐸 ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 ) )
42	14	fneq2i	⊢ ( 𝑄 Fn 𝐸 ↔ 𝑄 Fn ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
43	40 42	sylibr	⊢ ( 𝜒 → 𝑄 Fn 𝐸 )
44	1 2 3 4 5 6 7 8 9 10 11 12 13 14	bnj1452	⊢ ( 𝜒 → 𝐸 ∈ 𝐵 )
45	1 2 3 4 5 6 7 8 9 10 11 12 13 14 32 41 43 44	bnj1463	⊢ ( 𝜒 → 𝑄 ∈ 𝐶 )
46	45 38	jca	⊢ ( 𝜒 → ( 𝑄 ∈ 𝐶 ∧ dom 𝑄 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
47	1 2 3 4 5 6 7 8 9 10 11 12 46	bnj1491	⊢ ( ( 𝜒 ∧ 𝑄 ∈ V ) → ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
48	32 47	mpdan	⊢ ( 𝜒 → ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
49	48 4	bnj1198	⊢ ( 𝜒 → ∃ 𝑓 𝜏 )
50	31 49	nsyl3	⊢ ( 𝜒 → ¬ 𝑥 ∈ 𝐷 )
51	29 30 50	bnj1304	⊢ ¬ 𝜓
52	6 51	bnj1541	⊢ ( 𝑅 FrSe 𝐴 → 𝐷 = ∅ )
53	5 52	bnj1476	⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 ∃ 𝑓 𝜏 )
54	4	exbii	⊢ ( ∃ 𝑓 𝜏 ↔ ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
55		df-rex	⊢ ( ∃ 𝑓 ∈ 𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ↔ ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
56	54 55	bitr4i	⊢ ( ∃ 𝑓 𝜏 ↔ ∃ 𝑓 ∈ 𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
57	56	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑓 𝜏 ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑓 ∈ 𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
58	53 57	sylib	⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 ∃ 𝑓 ∈ 𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )

Metamath Proof Explorer

Theorem bnj1312

Proof