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Theorem fint 6551
Description: Function into an intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Hypothesis
Ref Expression
fint.1 𝐵 ≠ ∅
Assertion
Ref Expression
fint (𝐹:𝐴 𝐵 ↔ ∀𝑥𝐵 𝐹:𝐴𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem fint
StepHypRef Expression
1 ssint 4883 . . . 4 (ran 𝐹 𝐵 ↔ ∀𝑥𝐵 ran 𝐹𝑥)
21anbi2i 622 . . 3 ((𝐹 Fn 𝐴 ∧ ran 𝐹 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐵 ran 𝐹𝑥))
3 fint.1 . . . 4 𝐵 ≠ ∅
4 r19.28zv 4442 . . . 4 (𝐵 ≠ ∅ → (∀𝑥𝐵 (𝐹 Fn 𝐴 ∧ ran 𝐹𝑥) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐵 ran 𝐹𝑥)))
53, 4ax-mp 5 . . 3 (∀𝑥𝐵 (𝐹 Fn 𝐴 ∧ ran 𝐹𝑥) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐵 ran 𝐹𝑥))
62, 5bitr4i 279 . 2 ((𝐹 Fn 𝐴 ∧ ran 𝐹 𝐵) ↔ ∀𝑥𝐵 (𝐹 Fn 𝐴 ∧ ran 𝐹𝑥))
7 df-f 6352 . 2 (𝐹:𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 𝐵))
8 df-f 6352 . . 3 (𝐹:𝐴𝑥 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝑥))
98ralbii 3162 . 2 (∀𝑥𝐵 𝐹:𝐴𝑥 ↔ ∀𝑥𝐵 (𝐹 Fn 𝐴 ∧ ran 𝐹𝑥))
106, 7, 93bitr4i 304 1 (𝐹:𝐴 𝐵 ↔ ∀𝑥𝐵 𝐹:𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wne 3013  wral 3135  wss 3933  c0 4288   cint 4867  ran crn 5549   Fn wfn 6343  wf 6344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-v 3494  df-dif 3936  df-in 3940  df-ss 3949  df-nul 4289  df-int 4868  df-f 6352
This theorem is referenced by:  chintcli  29035
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