Theorem bnj1234

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	bnj1234.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1234.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1234.4	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1234.5	⊢ 𝐷 = { 𝑔 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑔 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑍 ) ) }
Assertion	bnj1234	⊢ 𝐶 = 𝐷

Proof

Step	Hyp	Ref	Expression
1		bnj1234.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
2		bnj1234.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
3		bnj1234.4	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
4		bnj1234.5	⊢ 𝐷 = { 𝑔 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑔 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑍 ) ) }
5		fneq1	⊢ ( 𝑓 = 𝑔 → ( 𝑓 Fn 𝑑 ↔ 𝑔 Fn 𝑑 ) )
6		fveq1	⊢ ( 𝑓 = 𝑔 → ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) )
7		reseq1	⊢ ( 𝑓 = 𝑔 → ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) = ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) )
8	7	opeq2d	⊢ ( 𝑓 = 𝑔 → ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ = ⟨ 𝑥 , ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ )
9	8 1 3	3eqtr4g	⊢ ( 𝑓 = 𝑔 → 𝑌 = 𝑍 )
10	9	fveq2d	⊢ ( 𝑓 = 𝑔 → ( 𝐺 ‘ 𝑌 ) = ( 𝐺 ‘ 𝑍 ) )
11	6 10	eqeq12d	⊢ ( 𝑓 = 𝑔 → ( ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ↔ ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑍 ) ) )
12	11	ralbidv	⊢ ( 𝑓 = 𝑔 → ( ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ↔ ∀ 𝑥 ∈ 𝑑 ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑍 ) ) )
13	5 12	anbi12d	⊢ ( 𝑓 = 𝑔 → ( ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) ↔ ( 𝑔 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑍 ) ) ) )
14	13	rexbidv	⊢ ( 𝑓 = 𝑔 → ( ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) ↔ ∃ 𝑑 ∈ 𝐵 ( 𝑔 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑍 ) ) ) )
15	14	cbvabv	⊢ { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) } = { 𝑔 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑔 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑍 ) ) }
16	15 2 4	3eqtr4i	⊢ 𝐶 = 𝐷