Theorem fint 5245

Description: Function into an intersection. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 14-Oct-1999.) (Revised by set.mm contributors, 18-Sep-2011.)

Hypothesis

Ref	Expression
fint.1	⊢ B ≠ ∅

Assertion

Ref	Expression
fint	⊢ (F:A–→∩B ↔ ∀x ∈ B F:A–→x)

Distinct variable groups:   x,A   x,B   x,F

Proof of Theorem fint

Step	Hyp	Ref	Expression
1		ssint 3942	. . . 4 ⊢ (ran F ⊆ ∩B ↔ ∀x ∈ B ran F ⊆ x)
2	1	anbi2i 675	. . 3 ⊢ ((F Fn A ∧ ran F ⊆ ∩B) ↔ (F Fn A ∧ ∀x ∈ B ran F ⊆ x))
3		fint.1	. . . 4 ⊢ B ≠ ∅
4		r19.28zv 3645	. . . 4 ⊢ (B ≠ ∅ → (∀x ∈ B (F Fn A ∧ ran F ⊆ x) ↔ (F Fn A ∧ ∀x ∈ B ran F ⊆ x)))
5	3, 4	ax-mp 5	. . 3 ⊢ (∀x ∈ B (F Fn A ∧ ran F ⊆ x) ↔ (F Fn A ∧ ∀x ∈ B ran F ⊆ x))
6	2, 5	bitr4i 243	. 2 ⊢ ((F Fn A ∧ ran F ⊆ ∩B) ↔ ∀x ∈ B (F Fn A ∧ ran F ⊆ x))
7		df-f 4791	. 2 ⊢ (F:A–→∩B ↔ (F Fn A ∧ ran F ⊆ ∩B))
8		df-f 4791	. . 3 ⊢ (F:A–→x ↔ (F Fn A ∧ ran F ⊆ x))
9	8	ralbii 2638	. 2 ⊢ (∀x ∈ B F:A–→x ↔ ∀x ∈ B (F Fn A ∧ ran F ⊆ x))
10	6, 7, 9	3bitr4i 268	1 ⊢ (F:A–→∩B ↔ ∀x ∈ B F:A–→x)

Syntax hints:   ↔ wb 176   ∧ wa 358   ≠ wne 2516  ∀wral 2614   ⊆ wss 3257  ∅c0 3550  ∩cint 3926  ran crn 4773   Fn wfn 4776  –→wf 4777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551  df-int 3927  df-f 4791

This theorem is referenced by: (None)