Theorem bnj1501

Description: Technical lemma for bnj1500 . 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	bnj1501.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1501.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1501.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1501.4	⊢ 𝐹 = ∪ 𝐶
	bnj1501.5	⊢ ( 𝜑 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) )
	bnj1501.6	⊢ ( 𝜓 ↔ ( 𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓 ) )
	bnj1501.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑑 ∈ 𝐵 ∧ dom 𝑓 = 𝑑 ) )
Assertion	bnj1501	⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1501.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1501.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1501.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1501.4	⊢ 𝐹 = ∪ 𝐶
5		bnj1501.5	⊢ ( 𝜑 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) )
6		bnj1501.6	⊢ ( 𝜓 ↔ ( 𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓 ) )
7		bnj1501.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑑 ∈ 𝐵 ∧ dom 𝑓 = 𝑑 ) )
8	5	simprbi	⊢ ( 𝜑 → 𝑥 ∈ 𝐴 )
9	1 2 3 4	bnj60	⊢ ( 𝑅 FrSe 𝐴 → 𝐹 Fn 𝐴 )
10	9	fndmd	⊢ ( 𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴 )
11	5 10	bnj832	⊢ ( 𝜑 → dom 𝐹 = 𝐴 )
12	8 11	eleqtrrd	⊢ ( 𝜑 → 𝑥 ∈ dom 𝐹 )
13	4	dmeqi	⊢ dom 𝐹 = dom ∪ 𝐶
14	3	bnj1317	⊢ ( 𝑤 ∈ 𝐶 → ∀ 𝑓 𝑤 ∈ 𝐶 )
15	14	bnj1400	⊢ dom ∪ 𝐶 = ∪ 𝑓 ∈ 𝐶 dom 𝑓
16	13 15	eqtri	⊢ dom 𝐹 = ∪ 𝑓 ∈ 𝐶 dom 𝑓
17	12 16	eleqtrdi	⊢ ( 𝜑 → 𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓 )
18	17	bnj1405	⊢ ( 𝜑 → ∃ 𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓 )
19	18 6	bnj1209	⊢ ( 𝜑 → ∃ 𝑓 𝜓 )
20	3	bnj1436	⊢ ( 𝑓 ∈ 𝐶 → ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) )
21	20	bnj1299	⊢ ( 𝑓 ∈ 𝐶 → ∃ 𝑑 ∈ 𝐵 𝑓 Fn 𝑑 )
22		fndm	⊢ ( 𝑓 Fn 𝑑 → dom 𝑓 = 𝑑 )
23	21 22	bnj31	⊢ ( 𝑓 ∈ 𝐶 → ∃ 𝑑 ∈ 𝐵 dom 𝑓 = 𝑑 )
24	6 23	bnj836	⊢ ( 𝜓 → ∃ 𝑑 ∈ 𝐵 dom 𝑓 = 𝑑 )
25	1 2 3 4 5 6	bnj1518	⊢ ( 𝜓 → ∀ 𝑑 𝜓 )
26	24 7 25	bnj1521	⊢ ( 𝜓 → ∃ 𝑑 𝜒 )
27	9	fnfund	⊢ ( 𝑅 FrSe 𝐴 → Fun 𝐹 )
28	5 27	bnj832	⊢ ( 𝜑 → Fun 𝐹 )
29	6 28	bnj835	⊢ ( 𝜓 → Fun 𝐹 )
30		elssuni	⊢ ( 𝑓 ∈ 𝐶 → 𝑓 ⊆ ∪ 𝐶 )
31	30 4	sseqtrrdi	⊢ ( 𝑓 ∈ 𝐶 → 𝑓 ⊆ 𝐹 )
32	6 31	bnj836	⊢ ( 𝜓 → 𝑓 ⊆ 𝐹 )
33	6	simp3bi	⊢ ( 𝜓 → 𝑥 ∈ dom 𝑓 )
34	29 32 33	bnj1502	⊢ ( 𝜓 → ( 𝐹 ‘ 𝑥 ) = ( 𝑓 ‘ 𝑥 ) )
35	1 2 3	bnj1514	⊢ ( 𝑓 ∈ 𝐶 → ∀ 𝑥 ∈ dom 𝑓 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) )
36	6 35	bnj836	⊢ ( 𝜓 → ∀ 𝑥 ∈ dom 𝑓 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) )
37	36 33	bnj1294	⊢ ( 𝜓 → ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) )
38	34 37	eqtrd	⊢ ( 𝜓 → ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) )
39	7 38	bnj835	⊢ ( 𝜒 → ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) )
40	7 29	bnj835	⊢ ( 𝜒 → Fun 𝐹 )
41	7 32	bnj835	⊢ ( 𝜒 → 𝑓 ⊆ 𝐹 )
42	1	bnj1517	⊢ ( 𝑑 ∈ 𝐵 → ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 )
43	7 42	bnj836	⊢ ( 𝜒 → ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 )
44	7 33	bnj835	⊢ ( 𝜒 → 𝑥 ∈ dom 𝑓 )
45	7	simp3bi	⊢ ( 𝜒 → dom 𝑓 = 𝑑 )
46	44 45	eleqtrd	⊢ ( 𝜒 → 𝑥 ∈ 𝑑 )
47	43 46	bnj1294	⊢ ( 𝜒 → pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 )
48	47 45	sseqtrrd	⊢ ( 𝜒 → pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ dom 𝑓 )
49	40 41 48	bnj1503	⊢ ( 𝜒 → ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) = ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) )
50	49	opeq2d	⊢ ( 𝜒 → ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ )
51	50 2	eqtr4di	⊢ ( 𝜒 → ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ = 𝑌 )
52	51	fveq2d	⊢ ( 𝜒 → ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) = ( 𝐺 ‘ 𝑌 ) )
53	39 52	eqtr4d	⊢ ( 𝜒 → ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) )
54	26 53	bnj593	⊢ ( 𝜓 → ∃ 𝑑 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) )
55	1 2 3 4	bnj1519	⊢ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) → ∀ 𝑑 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) )
56	54 55	bnj1397	⊢ ( 𝜓 → ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) )
57	19 56	bnj593	⊢ ( 𝜑 → ∃ 𝑓 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) )
58	1 2 3 4	bnj1520	⊢ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) → ∀ 𝑓 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) )
59	57 58	bnj1397	⊢ ( 𝜑 → ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) )
60	5 59	bnj1459	⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) )