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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–→Bx B F:A–→x)
Distinct variable groups:   x,A   x,B   x,F

Proof of Theorem fint
StepHypRef Expression
1 ssint 3942 . . . 4 (ran F Bx B ran F x)
21anbi2i 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)))
53, 4ax-mp 8 . . 3 (x B (F Fn A ran F x) ↔ (F Fn A x B ran F x))
62, 5bitr4i 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))
98ralbii 2638 . 2 (x B F:A–→xx B (F Fn A ran F x))
106, 7, 93bitr4i 268 1 (F:A–→Bx B F:A–→x)
 Colors of variables: wff setvar class 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-1 5  ax-2 6  ax-3 7  ax-mp 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)
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