Theorem bnj66

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	bnj66.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj66.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj66.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
Assertion	bnj66	⊢ ( 𝑔 ∈ 𝐶 → Rel 𝑔 )

Proof

Step	Hyp	Ref	Expression
1		bnj66.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj66.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj66.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		fneq1	⊢ ( 𝑔 = 𝑓 → ( 𝑔 Fn 𝑑 ↔ 𝑓 Fn 𝑑 ) )
5		fveq1	⊢ ( 𝑔 = 𝑓 → ( 𝑔 ‘ 𝑥 ) = ( 𝑓 ‘ 𝑥 ) )
6		reseq1	⊢ ( 𝑔 = 𝑓 → ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) = ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) )
7	6	opeq2d	⊢ ( 𝑔 = 𝑓 → ⟨ 𝑥 , ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ )
8	7 2	eqtr4di	⊢ ( 𝑔 = 𝑓 → ⟨ 𝑥 , ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ = 𝑌 )
9	8	fveq2d	⊢ ( 𝑔 = 𝑓 → ( 𝐺 ‘ ⟨ 𝑥 , ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) = ( 𝐺 ‘ 𝑌 ) )
10	5 9	eqeq12d	⊢ ( 𝑔 = 𝑓 → ( ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ↔ ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) )
11	10	ralbidv	⊢ ( 𝑔 = 𝑓 → ( ∀ 𝑥 ∈ 𝑑 ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ↔ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) )
12	4 11	anbi12d	⊢ ( 𝑔 = 𝑓 → ( ( 𝑔 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ) ↔ ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) ) )
13	12	rexbidv	⊢ ( 𝑔 = 𝑓 → ( ∃ 𝑑 ∈ 𝐵 ( 𝑔 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ) ↔ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) ) )
14	13	cbvabv	⊢ { 𝑔 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑔 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ) } = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
15	3 14	eqtr4i	⊢ 𝐶 = { 𝑔 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑔 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ) }
16	15	bnj1436	⊢ ( 𝑔 ∈ 𝐶 → ∃ 𝑑 ∈ 𝐵 ( 𝑔 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ) )
17		bnj1239	⊢ ( ∃ 𝑑 ∈ 𝐵 ( 𝑔 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ) → ∃ 𝑑 ∈ 𝐵 𝑔 Fn 𝑑 )
18		fnrel	⊢ ( 𝑔 Fn 𝑑 → Rel 𝑔 )
19	18	rexlimivw	⊢ ( ∃ 𝑑 ∈ 𝐵 𝑔 Fn 𝑑 → Rel 𝑔 )
20	16 17 19	3syl	⊢ ( 𝑔 ∈ 𝐶 → Rel 𝑔 )