Theorem bnj1497

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	bnj1497.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1497.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1497.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
Assertion	bnj1497	⊢ ∀ 𝑔 ∈ 𝐶 Fun 𝑔

Proof

Step	Hyp	Ref	Expression
1		bnj1497.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1497.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1497.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4	3	bnj1317	⊢ ( 𝑔 ∈ 𝐶 → ∀ 𝑓 𝑔 ∈ 𝐶 )
5	4	nf5i	⊢ Ⅎ 𝑓 𝑔 ∈ 𝐶
6		nfv	⊢ Ⅎ 𝑓 Fun 𝑔
7	5 6	nfim	⊢ Ⅎ 𝑓 ( 𝑔 ∈ 𝐶 → Fun 𝑔 )
8		eleq1w	⊢ ( 𝑓 = 𝑔 → ( 𝑓 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶 ) )
9		funeq	⊢ ( 𝑓 = 𝑔 → ( Fun 𝑓 ↔ Fun 𝑔 ) )
10	8 9	imbi12d	⊢ ( 𝑓 = 𝑔 → ( ( 𝑓 ∈ 𝐶 → Fun 𝑓 ) ↔ ( 𝑔 ∈ 𝐶 → Fun 𝑔 ) ) )
11	3	bnj1436	⊢ ( 𝑓 ∈ 𝐶 → ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) )
12	11	bnj1299	⊢ ( 𝑓 ∈ 𝐶 → ∃ 𝑑 ∈ 𝐵 𝑓 Fn 𝑑 )
13		fnfun	⊢ ( 𝑓 Fn 𝑑 → Fun 𝑓 )
14	12 13	bnj31	⊢ ( 𝑓 ∈ 𝐶 → ∃ 𝑑 ∈ 𝐵 Fun 𝑓 )
15	14	bnj1265	⊢ ( 𝑓 ∈ 𝐶 → Fun 𝑓 )
16	7 10 15	chvarfv	⊢ ( 𝑔 ∈ 𝐶 → Fun 𝑔 )
17	16	rgen	⊢ ∀ 𝑔 ∈ 𝐶 Fun 𝑔