Metamath Proof Explorer


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 ∣ ( 𝑑𝐴 ∧ ∀ 𝑥𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
bnj1497.2 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
bnj1497.3 𝐶 = { 𝑓 ∣ ∃ 𝑑𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥𝑑 ( 𝑓𝑥 ) = ( 𝐺𝑌 ) ) }
Assertion bnj1497 𝑔𝐶 Fun 𝑔

Proof

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