Theorem fint 2769

Description: Function into an intersection.

Hypothesis

Ref	Expression
fint.1	⊢ ¬ B = ∅

Assertion

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

Distinct variable group(s):   x,A   x,B   x,F

Proof of Theorem fint

Step	Hyp	Ref	Expression
1		fint.1	. . . . 5 ⊢ ¬ B = ∅
2		r19.3rzv 1767	. . . . 5 ⊢ (¬ B = ∅ → (F Fn A ↔ ∀x ∈ B F Fn A))
3	1, 2	ax-mp 6	. . . 4 ⊢ (F Fn A ↔ ∀x ∈ B F Fn A)
4		ssint 1980	. . . 4 ⊢ (ran F ⊆ ∩B ↔ ∀x ∈ B ran F ⊆ x)
5	3, 4	anbi12i 369	. . 3 ⊢ ((F Fn A ∧ ran F ⊆ ∩B) ↔ (∀x ∈ B F Fn A ∧ ∀x ∈ B ran F ⊆ x))
6		r19.26 1289	. . 3 ⊢ (∀x ∈ B (F Fn A ∧ ran F ⊆ x) ↔ (∀x ∈ B F Fn A ∧ ∀x ∈ B ran F ⊆ x))
7	5, 6	bitr4 154	. 2 ⊢ ((F Fn A ∧ ran F ⊆ ∩B) ↔ ∀x ∈ B (F Fn A ∧ ran F ⊆ x))
8		df-f 2434	. 2 ⊢ (F:A–→∩B ↔ (F Fn A ∧ ran F ⊆ ∩B))
9		df-f 2434	. . 3 ⊢ (F:A–→x ↔ (F Fn A ∧ ran F ⊆ x))
10	9	biral 1223	. 2 ⊢ (∀x ∈ B F:A–→x ↔ ∀x ∈ B (F Fn A ∧ ran F ⊆ x))
11	7, 8, 10	3bitr4 158	1 ⊢ (F:A–→∩B ↔ ∀x ∈ B F:A–→x)

Syntax hints:  ¬ wn 1   ↔ wb 127   ∧ wa 196   = wceq 1091  ∀wral 1201   ⊆ wss 1487  ∅c0 1707  ∩cint 1965  ran crn 2411   Fn wfn 2417  –→wf 2418

This theorem is referenced by:  chintcl 5296

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708  df-int 1966  df-f 2434

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