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Theorem fint 6722
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 4926 . . . 4 (ran 𝐹 𝐵 ↔ ∀𝑥𝐵 ran 𝐹𝑥)
21anbi2i 624 . . 3 ((𝐹 Fn 𝐴 ∧ ran 𝐹 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐵 ran 𝐹𝑥))
3 fint.1 . . . 4 𝐵 ≠ ∅
4 r19.28zv 4459 . . . 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 278 . 2 ((𝐹 Fn 𝐴 ∧ ran 𝐹 𝐵) ↔ ∀𝑥𝐵 (𝐹 Fn 𝐴 ∧ ran 𝐹𝑥))
7 df-f 6501 . 2 (𝐹:𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 𝐵))
8 df-f 6501 . . 3 (𝐹:𝐴𝑥 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝑥))
98ralbii 3097 . 2 (∀𝑥𝐵 𝐹:𝐴𝑥 ↔ ∀𝑥𝐵 (𝐹 Fn 𝐴 ∧ ran 𝐹𝑥))
106, 7, 93bitr4i 303 1 (𝐹:𝐴 𝐵 ↔ ∀𝑥𝐵 𝐹:𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wne 2944  wral 3065  wss 3911  c0 4283   cint 4908  ran crn 5635   Fn wfn 6492  wf 6493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-v 3448  df-dif 3914  df-in 3918  df-ss 3928  df-nul 4284  df-int 4909  df-f 6501
This theorem is referenced by:  chintcli  30276
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