Theorem bnj1442

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	bnj1442.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1442.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1442.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1442.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
	bnj1442.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
	bnj1442.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
	bnj1442.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
	bnj1442.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
	bnj1442.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
	bnj1442.10	⊢ 𝑃 = ∪ 𝐻
	bnj1442.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1442.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
	bnj1442.13	⊢ 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
	bnj1442.14	⊢ 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
	bnj1442.15	⊢ ( 𝜒 → 𝑃 Fn trCl ( 𝑥 , 𝐴 , 𝑅 ) )
	bnj1442.16	⊢ ( 𝜒 → 𝑄 Fn ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
	bnj1442.17	⊢ ( 𝜃 ↔ ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) )
	bnj1442.18	⊢ ( 𝜂 ↔ ( 𝜃 ∧ 𝑧 ∈ { 𝑥 } ) )
Assertion	bnj1442	⊢ ( 𝜂 → ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1442.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1442.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1442.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1442.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5		bnj1442.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
6		bnj1442.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
7		bnj1442.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
8		bnj1442.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
9		bnj1442.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
10		bnj1442.10	⊢ 𝑃 = ∪ 𝐻
11		bnj1442.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
12		bnj1442.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
13		bnj1442.13	⊢ 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
14		bnj1442.14	⊢ 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
15		bnj1442.15	⊢ ( 𝜒 → 𝑃 Fn trCl ( 𝑥 , 𝐴 , 𝑅 ) )
16		bnj1442.16	⊢ ( 𝜒 → 𝑄 Fn ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
17		bnj1442.17	⊢ ( 𝜃 ↔ ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) )
18		bnj1442.18	⊢ ( 𝜂 ↔ ( 𝜃 ∧ 𝑧 ∈ { 𝑥 } ) )
19	16	bnj930	⊢ ( 𝜒 → Fun 𝑄 )
20		opex	⊢ ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ ∈ V
21	20	snid	⊢ ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ ∈ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ }
22		elun2	⊢ ( ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ ∈ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } → ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ ∈ ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } ) )
23	21 22	ax-mp	⊢ ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ ∈ ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
24	23 12	eleqtrri	⊢ ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ ∈ 𝑄
25		funopfv	⊢ ( Fun 𝑄 → ( ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ ∈ 𝑄 → ( 𝑄 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑍 ) ) )
26	19 24 25	mpisyl	⊢ ( 𝜒 → ( 𝑄 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑍 ) )
27	17 26	bnj832	⊢ ( 𝜃 → ( 𝑄 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑍 ) )
28	18 27	bnj832	⊢ ( 𝜂 → ( 𝑄 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑍 ) )
29		elsni	⊢ ( 𝑧 ∈ { 𝑥 } → 𝑧 = 𝑥 )
30	18 29	simplbiim	⊢ ( 𝜂 → 𝑧 = 𝑥 )
31	30	fveq2d	⊢ ( 𝜂 → ( 𝑄 ‘ 𝑧 ) = ( 𝑄 ‘ 𝑥 ) )
32		bnj602	⊢ ( 𝑧 = 𝑥 → pred ( 𝑧 , 𝐴 , 𝑅 ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
33	32	reseq2d	⊢ ( 𝑧 = 𝑥 → ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) = ( 𝑄 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) )
34	30 33	syl	⊢ ( 𝜂 → ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) = ( 𝑄 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) )
35	12	bnj931	⊢ 𝑃 ⊆ 𝑄
36	35	a1i	⊢ ( 𝜒 → 𝑃 ⊆ 𝑄 )
37	6	simplbi	⊢ ( 𝜓 → 𝑅 FrSe 𝐴 )
38	7 37	bnj835	⊢ ( 𝜒 → 𝑅 FrSe 𝐴 )
39	5 7	bnj1212	⊢ ( 𝜒 → 𝑥 ∈ 𝐴 )
40		bnj906	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
41	38 39 40	syl2anc	⊢ ( 𝜒 → pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
42	15	fndmd	⊢ ( 𝜒 → dom 𝑃 = trCl ( 𝑥 , 𝐴 , 𝑅 ) )
43	41 42	sseqtrrd	⊢ ( 𝜒 → pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ dom 𝑃 )
44	19 36 43	bnj1503	⊢ ( 𝜒 → ( 𝑄 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) = ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) )
45	17 44	bnj832	⊢ ( 𝜃 → ( 𝑄 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) = ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) )
46	18 45	bnj832	⊢ ( 𝜂 → ( 𝑄 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) = ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) )
47	34 46	eqtrd	⊢ ( 𝜂 → ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) = ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) )
48	30 47	opeq12d	⊢ ( 𝜂 → ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩ = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ )
49	48 13 11	3eqtr4g	⊢ ( 𝜂 → 𝑊 = 𝑍 )
50	49	fveq2d	⊢ ( 𝜂 → ( 𝐺 ‘ 𝑊 ) = ( 𝐺 ‘ 𝑍 ) )
51	28 31 50	3eqtr4d	⊢ ( 𝜂 → ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 ) )