ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f1ssr GIF version

Theorem f1ssr 5207
Description: Combine a one-to-one function with a restriction on the domain. (Contributed by Stefan O'Rear, 20-Feb-2015.)
Assertion
Ref Expression
f1ssr ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → 𝐹:𝐴1-1𝐶)

Proof of Theorem f1ssr
StepHypRef Expression
1 f1fn 5202 . . . 4 (𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
21adantr 270 . . 3 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → 𝐹 Fn 𝐴)
3 simpr 108 . . 3 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → ran 𝐹𝐶)
4 df-f 5006 . . 3 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶))
52, 3, 4sylanbrc 408 . 2 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → 𝐹:𝐴𝐶)
6 df-f1 5007 . . . 4 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
76simprbi 269 . . 3 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
87adantr 270 . 2 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → Fun 𝐹)
9 df-f1 5007 . 2 (𝐹:𝐴1-1𝐶 ↔ (𝐹:𝐴𝐶 ∧ Fun 𝐹))
105, 8, 9sylanbrc 408 1 ((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → 𝐹:𝐴1-1𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wss 2997  ccnv 4427  ran crn 4429  Fun wfun 4996   Fn wfn 4997  wf 4998  1-1wf1 4999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115  df-f 5006  df-f1 5007
This theorem is referenced by:  f1ff1  5208
  Copyright terms: Public domain W3C validator