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

Theorem respreima 5596
Description: The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
respreima (Fun 𝐹 → ((𝐹𝐵) “ 𝐴) = ((𝐹𝐴) ∩ 𝐵))

Proof of Theorem respreima
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 funfn 5201 . . 3 (Fun 𝐹𝐹 Fn dom 𝐹)
2 elin 3290 . . . . . . . . 9 (𝑥 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝑥𝐵𝑥 ∈ dom 𝐹))
3 ancom 264 . . . . . . . . 9 ((𝑥𝐵𝑥 ∈ dom 𝐹) ↔ (𝑥 ∈ dom 𝐹𝑥𝐵))
42, 3bitri 183 . . . . . . . 8 (𝑥 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝑥 ∈ dom 𝐹𝑥𝐵))
54anbi1i 454 . . . . . . 7 ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴))
6 fvres 5493 . . . . . . . . . 10 (𝑥𝐵 → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
76eleq1d 2226 . . . . . . . . 9 (𝑥𝐵 → (((𝐹𝐵)‘𝑥) ∈ 𝐴 ↔ (𝐹𝑥) ∈ 𝐴))
87adantl 275 . . . . . . . 8 ((𝑥 ∈ dom 𝐹𝑥𝐵) → (((𝐹𝐵)‘𝑥) ∈ 𝐴 ↔ (𝐹𝑥) ∈ 𝐴))
98pm5.32i 450 . . . . . . 7 (((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ (𝐹𝑥) ∈ 𝐴))
105, 9bitri 183 . . . . . 6 ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ (𝐹𝑥) ∈ 𝐴))
1110a1i 9 . . . . 5 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ (𝐹𝑥) ∈ 𝐴)))
12 an32 552 . . . . 5 (((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ (𝐹𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∧ 𝑥𝐵))
1311, 12bitrdi 195 . . . 4 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∧ 𝑥𝐵)))
14 fnfun 5268 . . . . . . . 8 (𝐹 Fn dom 𝐹 → Fun 𝐹)
15 funres 5212 . . . . . . . 8 (Fun 𝐹 → Fun (𝐹𝐵))
1614, 15syl 14 . . . . . . 7 (𝐹 Fn dom 𝐹 → Fun (𝐹𝐵))
17 dmres 4888 . . . . . . 7 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
1816, 17jctir 311 . . . . . 6 (𝐹 Fn dom 𝐹 → (Fun (𝐹𝐵) ∧ dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)))
19 df-fn 5174 . . . . . 6 ((𝐹𝐵) Fn (𝐵 ∩ dom 𝐹) ↔ (Fun (𝐹𝐵) ∧ dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)))
2018, 19sylibr 133 . . . . 5 (𝐹 Fn dom 𝐹 → (𝐹𝐵) Fn (𝐵 ∩ dom 𝐹))
21 elpreima 5587 . . . . 5 ((𝐹𝐵) Fn (𝐵 ∩ dom 𝐹) → (𝑥 ∈ ((𝐹𝐵) “ 𝐴) ↔ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴)))
2220, 21syl 14 . . . 4 (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((𝐹𝐵) “ 𝐴) ↔ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴)))
23 elin 3290 . . . . 5 (𝑥 ∈ ((𝐹𝐴) ∩ 𝐵) ↔ (𝑥 ∈ (𝐹𝐴) ∧ 𝑥𝐵))
24 elpreima 5587 . . . . . 6 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹𝐴) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴)))
2524anbi1d 461 . . . . 5 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐹𝐴) ∧ 𝑥𝐵) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∧ 𝑥𝐵)))
2623, 25syl5bb 191 . . . 4 (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((𝐹𝐴) ∩ 𝐵) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∧ 𝑥𝐵)))
2713, 22, 263bitr4d 219 . . 3 (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((𝐹𝐵) “ 𝐴) ↔ 𝑥 ∈ ((𝐹𝐴) ∩ 𝐵)))
281, 27sylbi 120 . 2 (Fun 𝐹 → (𝑥 ∈ ((𝐹𝐵) “ 𝐴) ↔ 𝑥 ∈ ((𝐹𝐴) ∩ 𝐵)))
2928eqrdv 2155 1 (Fun 𝐹 → ((𝐹𝐵) “ 𝐴) = ((𝐹𝐴) ∩ 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1335  wcel 2128  cin 3101  ccnv 4586  dom cdm 4587  cres 4589  cima 4590  Fun wfun 5165   Fn wfn 5166  cfv 5171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-fv 5179
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator