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Theorem fint 6761
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 4959 . . . 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 4493 . . . 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 6538 . 2 (𝐹:𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 𝐵))
8 df-f 6538 . . 3 (𝐹:𝐴𝑥 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝑥))
98ralbii 3085 . 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 395  wne 2932  wral 3053  wss 3941  c0 4315   cint 4941  ran crn 5668   Fn wfn 6529  wf 6530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-v 3468  df-dif 3944  df-in 3948  df-ss 3958  df-nul 4316  df-int 4942  df-f 6538
This theorem is referenced by:  chintcli  31056
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