MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  respreima Structured version   Visualization version   GIF version

Theorem respreima 7085
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 6597 . . 3 (Fun 𝐹𝐹 Fn dom 𝐹)
2 elin 3978 . . . . . . . . 9 (𝑥 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝑥𝐵𝑥 ∈ dom 𝐹))
32biancomi 462 . . . . . . . 8 (𝑥 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝑥 ∈ dom 𝐹𝑥𝐵))
43anbi1i 624 . . . . . . 7 ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴))
5 fvres 6925 . . . . . . . . . 10 (𝑥𝐵 → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
65eleq1d 2823 . . . . . . . . 9 (𝑥𝐵 → (((𝐹𝐵)‘𝑥) ∈ 𝐴 ↔ (𝐹𝑥) ∈ 𝐴))
76adantl 481 . . . . . . . 8 ((𝑥 ∈ dom 𝐹𝑥𝐵) → (((𝐹𝐵)‘𝑥) ∈ 𝐴 ↔ (𝐹𝑥) ∈ 𝐴))
87pm5.32i 574 . . . . . . 7 (((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ (𝐹𝑥) ∈ 𝐴))
94, 8bitri 275 . . . . . 6 ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ (𝐹𝑥) ∈ 𝐴))
109a1i 11 . . . . 5 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ (𝐹𝑥) ∈ 𝐴)))
11 an32 646 . . . . 5 (((𝑥 ∈ dom 𝐹𝑥𝐵) ∧ (𝐹𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∧ 𝑥𝐵))
1210, 11bitrdi 287 . . . 4 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∧ 𝑥𝐵)))
13 fnfun 6668 . . . . . . 7 (𝐹 Fn dom 𝐹 → Fun 𝐹)
1413funresd 6610 . . . . . 6 (𝐹 Fn dom 𝐹 → Fun (𝐹𝐵))
15 dmres 6031 . . . . . 6 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
16 df-fn 6565 . . . . . 6 ((𝐹𝐵) Fn (𝐵 ∩ dom 𝐹) ↔ (Fun (𝐹𝐵) ∧ dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)))
1714, 15, 16sylanblrc 590 . . . . 5 (𝐹 Fn dom 𝐹 → (𝐹𝐵) Fn (𝐵 ∩ dom 𝐹))
18 elpreima 7077 . . . . 5 ((𝐹𝐵) Fn (𝐵 ∩ dom 𝐹) → (𝑥 ∈ ((𝐹𝐵) “ 𝐴) ↔ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴)))
1917, 18syl 17 . . . 4 (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((𝐹𝐵) “ 𝐴) ↔ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹𝐵)‘𝑥) ∈ 𝐴)))
20 elin 3978 . . . . 5 (𝑥 ∈ ((𝐹𝐴) ∩ 𝐵) ↔ (𝑥 ∈ (𝐹𝐴) ∧ 𝑥𝐵))
21 elpreima 7077 . . . . . 6 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹𝐴) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴)))
2221anbi1d 631 . . . . 5 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐹𝐴) ∧ 𝑥𝐵) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∧ 𝑥𝐵)))
2320, 22bitrid 283 . . . 4 (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((𝐹𝐴) ∩ 𝐵) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∧ 𝑥𝐵)))
2412, 19, 233bitr4d 311 . . 3 (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((𝐹𝐵) “ 𝐴) ↔ 𝑥 ∈ ((𝐹𝐴) ∩ 𝐵)))
251, 24sylbi 217 . 2 (Fun 𝐹 → (𝑥 ∈ ((𝐹𝐵) “ 𝐴) ↔ 𝑥 ∈ ((𝐹𝐴) ∩ 𝐵)))
2625eqrdv 2732 1 (Fun 𝐹 → ((𝐹𝐵) “ 𝐴) = ((𝐹𝐴) ∩ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  cin 3961  ccnv 5687  dom cdm 5688  cres 5690  cima 5691  Fun wfun 6556   Fn wfn 6557  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570
This theorem is referenced by:  paste  23317  restmetu  24598  eulerpartlemt  34352  smfres  46745
  Copyright terms: Public domain W3C validator