Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  fintm Structured version   GIF version

Theorem fintm 5016
 Description: Function into an intersection. (Contributed by Jim Kingdon, 28-Dec-2018.)
Hypothesis
Ref Expression
fintm.1 x x B
Assertion
Ref Expression
fintm (𝐹:A Bx B 𝐹:Ax)
Distinct variable groups:   x,A   x,B   x,𝐹

Proof of Theorem fintm
StepHypRef Expression
1 ssint 3621 . . . 4 (ran 𝐹 Bx B ran 𝐹x)
21anbi2i 430 . . 3 ((𝐹 Fn A ran 𝐹 B) ↔ (𝐹 Fn A x B ran 𝐹x))
3 fintm.1 . . . 4 x x B
4 r19.28mv 3308 . . . 4 (x x B → (x B (𝐹 Fn A ran 𝐹x) ↔ (𝐹 Fn A x B ran 𝐹x)))
53, 4ax-mp 7 . . 3 (x B (𝐹 Fn A ran 𝐹x) ↔ (𝐹 Fn A x B ran 𝐹x))
62, 5bitr4i 176 . 2 ((𝐹 Fn A ran 𝐹 B) ↔ x B (𝐹 Fn A ran 𝐹x))
7 df-f 4848 . 2 (𝐹:A B ↔ (𝐹 Fn A ran 𝐹 B))
8 df-f 4848 . . 3 (𝐹:Ax ↔ (𝐹 Fn A ran 𝐹x))
98ralbii 2324 . 2 (x B 𝐹:Axx B (𝐹 Fn A ran 𝐹x))
106, 7, 93bitr4i 201 1 (𝐹:A Bx B 𝐹:Ax)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∃wex 1378   ∈ wcel 1390  ∀wral 2300   ⊆ wss 2911  ∩ cint 3605  ran crn 4288   Fn wfn 4839  ⟶wf 4840 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-int 3606  df-f 4848 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator