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

Theorem fintm 4996
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 3601 . . . 4 (ran 𝐹 Bx B ran 𝐹x)
21anbi2i 433 . . 3 ((𝐹 Fn A ran 𝐹 B) ↔ (𝐹 Fn A x B ran 𝐹x))
3 fintm.1 . . . 4 x x B
4 r19.28mv 3289 . . . 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 4829 . 2 (𝐹:A B ↔ (𝐹 Fn A ran 𝐹 B))
8 df-f 4829 . . 3 (𝐹:Ax ↔ (𝐹 Fn A ran 𝐹x))
98ralbii 2304 . 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 1358   wcel 1370  wral 2280  wss 2890   cint 3585  ran crn 4269   Fn wfn 4820  wf 4821
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-v 2533  df-in 2897  df-ss 2904  df-int 3586  df-f 4829
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator